From f75950a7d9437628010eec55b024281092fc6bdd Mon Sep 17 00:00:00 2001 From: Paul Baksic Date: Wed, 29 Jul 2026 18:00:25 +0200 Subject: [PATCH 01/10] start generic changes for main principles --- .../10_Integration_Scheme.md | 45 ++++++++++++------- 1 file changed, 30 insertions(+), 15 deletions(-) diff --git a/20_Simulation_Principles/40_System_Resolution/10_Integration_Scheme.md b/20_Simulation_Principles/40_System_Resolution/10_Integration_Scheme.md index 3b45455d9..2fd256a7d 100644 --- a/20_Simulation_Principles/40_System_Resolution/10_Integration_Scheme.md +++ b/20_Simulation_Principles/40_System_Resolution/10_Integration_Scheme.md @@ -1,9 +1,9 @@ Integration Schemes =================== -All dynamic simulations assume to discretize the temporal evolution of the system through small time steps. This time step is usually noted *dt*. An integration scheme is the [numerical method](https://en.wikipedia.org/wiki/Numerical_methods_for_ordinary_differential_equations) describing how to find the approximate solution for ordinary differential equations (ODE). +All dynamic simulations assume to discretize the temporal evolution of the system through small time steps. This time step is usually noted *dt*. An integration scheme is the [numerical method](https://en.wikipedia.org/wiki/Numerical_methods_for_ordinary_differential_equations) describing how to linearly relate the different time derivatives in order to discretize and linearize those ODE. -They are usually called **ODESolver** in SOFA. +They are usually called **IntegrationScheme** in SOFA. Let's write our ordinary differential equation of a function *y* as follows: @@ -11,7 +11,7 @@ $$ \frac{dy}{dt}=f\left( t,y(t)\right) $$ -ODESolver defines how to go from the current time step (t) to the next (t + dt), which will structure the linear system $\mathbf{A}x=b$. The integration scheme therefore defines which forces impact the left hand side matrix $\mathbf{A}$ and which forces contribute to the right hand side vector *b*: +IntegrationScheme defines how to go from the current time step (t) to the next (t + dt), which will structure the linear system $\mathbf{A}x=b$. The integration scheme therefore defines which forces impact the left hand side matrix $\mathbf{A}$ and which forces contribute to the right hand side vector *b*: - explicit contributions depending on the degrees of freedom (DOFs) at the current time step $x(t)$ will contribute to the $b$ vector - while implicit contributions depending on the degrees of freedom (DOFs) at the next step $x(t+dt)$ (unknown) will contribute to $\mathbf{A}$. @@ -38,11 +38,11 @@ $$ Explicit schemes are usually known as being fast to solve (since the created linear system is lighter) but they require very small time steps, unless they may undergo stability issues. They are known to efficiently solve non-stiff problems. -Explicit ODESolvers in SOFA: +Explicit IntegrationScheme in SOFA: -- [EulerExplicitSolver](../../../components/odesolver/forward/eulerexplicitsolver/) -- [CentralDifferenceSolver](../../../components/odesolver/forward/centraldifferencesolver/) -- [RungeKutta2Solver](../../../components/odesolver/forward/rungekutta2solver/) +- [EulerExplicitIntegrationScheme](../../../components/integrationscheme/forward/eulerexplicitintegrationscheme/) +- [CentralDifferenceIntegrationScheme](../../../components/integrationscheme/forward/centraldifferenceintegrationscheme/) +- [RungeKutta2IntegrationScheme](../../../components/integrationscheme/forward/rungekutta2integrationscheme/) ### Implicit scheme @@ -61,17 +61,32 @@ $$ Implicit schemes are known as being slower to solve (the outcoming linear system is more complex) but they are way more stable than explicit schemes. Stiff differential equations require the use of implicit schemes. -Implicit ODESolvers in SOFA: +Implicit IntegrationScheme in SOFA: + +- [EulerImplicitIntegrationScheme](../../../components/integrationscheme/backward/eulerimplicitintegrationscheme/) +- [NewmarkImplicitIntegrationScheme](../../../components/integrationscheme/backward/newmarkimplicitintegrationscheme/) +- [BDFIntegrationScheme](../../../components/integrationscheme/backward/variationalsymplecticintegrationscheme/) + + +Solving for non linearities +---------------- + +As it has been seen, integrating through time boils down to building and solving a linear system. What is hidden behind this is that the mechanics needs to be linearized to be summarized in a linear system. While has no effect for linear elasticity, it can result in pretty bad dynamics in the case of hyperelasticity. + +For explicit integration scheme, there is no strategy other than reducing the timestep to try to improve such non-linearities. But, because of the nature of implicit integration scheme, one can take advantage of using a non-linear solver to compute the integration in order to better take into account the non-linearities. + +The current design of Implicit integration scheme is based on this finding to enable the use of Newton-Raphson solver at the level of the simulation to compute dynamics. + + + -- [EulerImplicitSolver](../../../components/odesolver/backward/eulerimplicitsolver/) -- [NewmarkImplicitSolver](../../../components/odesolver/backward/newmarkimplicitsolver/) -- [VariationalSymplecticSolver](../../../components/odesolver/backward/variationalsymplecticsolver/) In the SOFA code ---------------- +//TODO -The integration scheme is described in the `solve()` function of the ODESolver. This *solve()* function is called by the [AnimationLoop](../../animation-loop/) (through a dedicated visitor) and builds the complete linear system $\mathbf{A}x=b$. +The integration scheme is described in the `integrate()` function of the IntegrationScheme. This *integrate()* function is called by the [AnimationLoop](../../animation-loop/) (through a dedicated visitor) and builds the complete linear system $\mathbf{A}x=b$. ### Specification of the scheme @@ -107,9 +122,9 @@ Again, Depending on the scheme (explicit or implicit, see previous paragraph), t -### State vectors in ODESolver +### State vectors in IntegrationScheme -In order to build the linear matrix system, the ODESolver uses information contained in [state vectors](../../mechanicalobject/#state-vectors) (like DOFs and their derivatives) within the scope of the ODESolver. The ODESolver does not access the state vectors directly. It accesses the state vectors remotely using visitors, which traverse the graph starting from the node which contains the solver. This keeps the implementation of the solver independent of the simulated objects and their types. +In order to build the linear matrix system, the IntegrationScheme uses information contained in [state vectors](../../mechanicalobject/#state-vectors) (like DOFs and their derivatives) within the scope of the IntegrationScheme. The IntegrationScheme does not access the state vectors directly. It accesses the state vectors remotely using visitors, which traverse the graph starting from the node which contains the solver. This keeps the implementation of the solver independent of the simulated objects and their types. Each type of solver may use different auxiliary state vectors to implement their simulation method. State vectors (MultiVec) are allocated and processed in the scope of the solver in a thread-safe way using an instance of _simulation::common::VectorOperations_. For instance, a Runge-Kutta algorithms needs to save the result of previous time steps. @@ -125,7 +140,7 @@ MultiVecCoord previousPos(&vop, previousPosID); // additional vector ### Compute the solution -In most cases, the matrix system $\mathbf{A}x=b$ can then be sent to a [LinearSolver](../../system-resolution/linear-solver/) in charge of finally solving the system defined according to the chosen scheme. Within the function *ODESolver::solve()*, the call to the LinearSolver will appear through the function call: +In most cases, the matrix system $\mathbf{A}x=b$ can then be sent to a [LinearSolver](../../system-resolution/linear-solver/) in charge of finally solving the system defined according to the chosen scheme. Within the function *IntegrationScheme::solve()*, the call to the LinearSolver will appear through the function call: ``` cpp matrix.solve(x, b); From c7cfaeb0f52b4f7202d648a70977f1a43660300e Mon Sep 17 00:00:00 2001 From: Paul Baksic Date: Wed, 29 Jul 2026 18:03:32 +0200 Subject: [PATCH 02/10] Add some equations --- .../10_Integration_Scheme.md | 23 ++++++++++++++++++- 1 file changed, 22 insertions(+), 1 deletion(-) diff --git a/20_Simulation_Principles/40_System_Resolution/10_Integration_Scheme.md b/20_Simulation_Principles/40_System_Resolution/10_Integration_Scheme.md index 2fd256a7d..05d11a9f7 100644 --- a/20_Simulation_Principles/40_System_Resolution/10_Integration_Scheme.md +++ b/20_Simulation_Principles/40_System_Resolution/10_Integration_Scheme.md @@ -75,10 +75,31 @@ As it has been seen, integrating through time boils down to building and solving For explicit integration scheme, there is no strategy other than reducing the timestep to try to improve such non-linearities. But, because of the nature of implicit integration scheme, one can take advantage of using a non-linear solver to compute the integration in order to better take into account the non-linearities. -The current design of Implicit integration scheme is based on this finding to enable the use of Newton-Raphson solver at the level of the simulation to compute dynamics. +The current design of Implicit integration scheme is based on this finding to enable the use of Newton-Raphson solver at the level of the simulation to compute dynamics. But to present it let's first dive into what it takes to use a Newton-Raphson solver : we need to express the time step as a root-finding problem. +### From Dynamic equations to root finding problem +First let's introduce the dynamic equations we wish to solve. +$$ +\boldsymbol{M}\boldsymbol{a} = \mathcal{F}(\boldsymbol{x},\boldsymbol{v}) + \boldsymbol{F_{\text{ext}}} +$$ + + +$$ +\begin{aligned} +g_{\boldsymbol{x}}^{(t,h)} &: (\boldsymbol{v}_{t+h}, \boldsymbol{a}_{t+h}) &\mapsto \boldsymbol{x}_{t+h} \\ +g_{\boldsymbol{v}}^{(t,h)} &: (\boldsymbol{a}_{t+h}) &\mapsto \boldsymbol{v}_{t+h} +\end{aligned} +$$ + +$$ +\begin{cases} +\boldsymbol{M}\boldsymbol{a}_{t+h} &= \mathcal{F}(\boldsymbol{x}_{t+h},\boldsymbol{v}_{t+h}) + \boldsymbol{F_{\text{ext}}} \\ +\boldsymbol{x}_{t+h} &= g_{\boldsymbol{x}}^{(t,h)}(\boldsymbol{v}_{t+h}, \boldsymbol{a}_{t+h}) \\ +\boldsymbol{v}_{t+h} &= g_{\boldsymbol{v}}^{(t,h)}(\boldsymbol{a}_{t+h}) +\end{cases} +$$ From 55b9c777d2da6d7da3cbdd862422ce91cb32cfa9 Mon Sep 17 00:00:00 2001 From: Paul Baksic Date: Thu, 30 Jul 2026 14:57:33 +0200 Subject: [PATCH 03/10] Add all genberal equations --- .../10_Integration_Scheme.md | 265 ++++++++++++++---- 1 file changed, 211 insertions(+), 54 deletions(-) diff --git a/20_Simulation_Principles/40_System_Resolution/10_Integration_Scheme.md b/20_Simulation_Principles/40_System_Resolution/10_Integration_Scheme.md index 05d11a9f7..6edab3a99 100644 --- a/20_Simulation_Principles/40_System_Resolution/10_Integration_Scheme.md +++ b/20_Simulation_Principles/40_System_Resolution/10_Integration_Scheme.md @@ -71,115 +71,272 @@ Implicit IntegrationScheme in SOFA: Solving for non linearities ---------------- -As it has been seen, integrating through time boils down to building and solving a linear system. What is hidden behind this is that the mechanics needs to be linearized to be summarized in a linear system. While has no effect for linear elasticity, it can result in pretty bad dynamics in the case of hyperelasticity. +As it has been seen, integrating through time boils down to building and solving a linear system. What is hidden behind this is that the mechanics needs to be linearized to be summarized in a linear system. While it has no effect for linear elasticity, it can result in pretty bad dynamics in the case of hyperelasticity. For explicit integration scheme, there is no strategy other than reducing the timestep to try to improve such non-linearities. But, because of the nature of implicit integration scheme, one can take advantage of using a non-linear solver to compute the integration in order to better take into account the non-linearities. The current design of Implicit integration scheme is based on this finding to enable the use of Newton-Raphson solver at the level of the simulation to compute dynamics. But to present it let's first dive into what it takes to use a Newton-Raphson solver : we need to express the time step as a root-finding problem. -### From Dynamic equations to root finding problem +> **Note :**\ +> Currently, SOFA doesn't propose any Newton-Raphson algorithm able to solve for non-linear mechanics. It is an ongoing work that'll be release soon. But the refactoring of the ODE solver has been made in order to enable such new solver to be implemented. It is thus important to explain the design choices at the light of future development. + +### 1) From Dynamic equations to root finding problem First let's introduce the dynamic equations we wish to solve. $$ \boldsymbol{M}\boldsymbol{a} = \mathcal{F}(\boldsymbol{x},\boldsymbol{v}) + \boldsymbol{F_{\text{ext}}} +\tag{1.1} $$ +With $\boldsymbol{a}$ the acceleration, $\boldsymbol{M}$ the mass matrix and $\mathcal{F}(\boldsymbol{x},\boldsymbol{v})$ being a non linear function of the position ($\boldsymbol{x}$) and velocity ($\boldsymbol{v}$). +Now let's consider that we have the solution to this equation at a time $t$. This means having the tensor $\boldsymbol{X}_t = (\boldsymbol{x}_t, \boldsymbol{v}_t, \boldsymbol{a}_t)$ that satisfy @eq:eq-dyn. In order to integrate the unknown along time and know their value at a future timestamp $t + h$, we need to introduce a relationship between the three unknown. Indeed, with only one equation but three unknown (position, velocity and acceleration), we need to introduce more equations to get a solvable system. This set of new equations is exactly the _integration scheme_ and can be defined as the following functions. $$ + \begin{aligned} g_{\boldsymbol{x}}^{(t,h)} &: (\boldsymbol{v}_{t+h}, \boldsymbol{a}_{t+h}) &\mapsto \boldsymbol{x}_{t+h} \\ g_{\boldsymbol{v}}^{(t,h)} &: (\boldsymbol{a}_{t+h}) &\mapsto \boldsymbol{v}_{t+h} +\tag{1.2} \end{aligned} $$ +Now one can rewrite the set of equations to be solved : + $$ \begin{cases} \boldsymbol{M}\boldsymbol{a}_{t+h} &= \mathcal{F}(\boldsymbol{x}_{t+h},\boldsymbol{v}_{t+h}) + \boldsymbol{F_{\text{ext}}} \\ \boldsymbol{x}_{t+h} &= g_{\boldsymbol{x}}^{(t,h)}(\boldsymbol{v}_{t+h}, \boldsymbol{a}_{t+h}) \\ \boldsymbol{v}_{t+h} &= g_{\boldsymbol{v}}^{(t,h)}(\boldsymbol{a}_{t+h}) \end{cases} +\tag{1.3} $$ +With the unknown $\boldsymbol{X}_{t+h} = (\boldsymbol{x}_{t+h}, \boldsymbol{v}_{t+h}, \boldsymbol{a}_{t+h})$. +#### 1.1) General form +Let define a function $\mathcal{G}_{t} (\boldsymbol{X}_{t+h} = (\boldsymbol{x}_{t+h}, \boldsymbol{v}_{t+h}, \boldsymbol{a}_{t+h}))$ such as its root is the solution of the set of equation (1.3): +$$ +\mathcal{G}_{t}(\boldsymbol{X} = (\boldsymbol{x}, \boldsymbol{v}, \boldsymbol{a}))) = +\begin{cases} +\boldsymbol{M}\boldsymbol{a} - \mathcal{F}(\boldsymbol{x},\boldsymbol{v}) - \boldsymbol{F_{\text{ext}}} \\ +\boldsymbol{x} - g_{\boldsymbol{x}}^{(t,h)}(\boldsymbol{v}, \boldsymbol{a}) \\ +\boldsymbol{v} - g_{\boldsymbol{v}}^{(t,h)}(\boldsymbol{a}) +\end{cases} +\tag{1.4} +$$ +We will see later that using this general form ends up in solving for the difference in acceleration in one timestep. We will call the family of integration scheme using this forme the **_Velocity-based integration schemes_**. +#### 1.2) Simplified form +To simplify this set of equation, if $g_v^{(t,h)}$ is invertible, one can rewrite the set as : +$$ +\begin{cases} +\boldsymbol{M}g_{\boldsymbol{v}}^{(t,h)-1}(\boldsymbol{v}_{t+h}) &= \mathcal{F}(\boldsymbol{x}_{t+h},\boldsymbol{v}_{t+h}) + \boldsymbol{F_{\text{ext}}} \\ +\boldsymbol{x}_{t+h} &= \tilde{g}_{\boldsymbol{x}}^{(t,h)}(\boldsymbol{v}_{t+h}) +\end{cases} +\tag{1.5} +$$ +With $\tilde{g}_{\boldsymbol{x}}^{(t,h)}(\boldsymbol{v}_{t+h}) = g_{\boldsymbol{x}}^{(t,h)}(\boldsymbol{v}_{t+h}, g_v^{(t,h)-1}(\boldsymbol{v}_{t+h}))$. We will use this simplified form for the following equations. +Let define a function $\mathcal{G}_{t} (\tilde{\boldsymbol{X}}_{t+h} = (\boldsymbol{x}_{t+h}, \boldsymbol{v}_{t+h}))$ such as its root is still the solution of (1.3): +$$ +\mathcal{G}_{t}(\tilde{\boldsymbol{X}} = (\boldsymbol{x}, \boldsymbol{v})) = +\begin{cases} +\boldsymbol{M}g_v^{(t,h)-1}(\boldsymbol{v}) - \mathcal{F}(\boldsymbol{x},\boldsymbol{v}) - \boldsymbol{F_{\text{ext}}} \\ +\boldsymbol{x} - \tilde{g}_{x}^{(t,h)}(\boldsymbol{v}) +\end{cases} +\tag{1.6} +$$ +To find the root of such non-linear function, one way to do it is to use a non-linear root finder, the most common one is the Newton-Raphson algorithm -In the SOFA code ----------------- -//TODO - -The integration scheme is described in the `integrate()` function of the IntegrationScheme. This *integrate()* function is called by the [AnimationLoop](../../animation-loop/) (through a dedicated visitor) and builds the complete linear system $\mathbf{A}x=b$. - - -### Specification of the scheme -The construction of the linear system changes whether the integration scheme is explicit or implicit, which is specified by: +### 2) Newton-Raphson equations +#### 2.1) Basics on Newton-Raphson -- for explicit cases -``` cpp -mop->setImplicit(false); -``` -- for implicit cases -``` cpp -mop->setImplicit(true); -``` +The Newton-Raphson algorithm is an iterative algorithm which goal is to find the root of a non-linear function that is at least $\mathscr{C}^1$ and has values in $\mathbb{R}^p$. For the sake of simplicity we will consider $n = p$ for the rest of this document (the main difference lying in the invertibility of the Jacobian of the function, when $n \neq p$ or when the Jacobian is not invertible a pseudo inverse must be used). +Let consider a $\mathscr{C}^1$ non-linear function $\mathcal{G} : \mathbb{R}^n \mapsto \mathbb{R}^n$. Given a current guess $\boldsymbol{X}^{(i)}$, the goal of the algorithm is to find $\mathrm{d}\boldsymbol{X}^{(i)}$ such as +$$ +\mathcal{G}(\boldsymbol{X}^{(i+1)}) = \mathcal{G}(\boldsymbol{X}^{(i)} + \mathrm{d}\boldsymbol{X}^{(i)}) = 0 +$$ +To do this the function $\mathcal{G}$ is linearized at the current guess to express the previous equation as a linear equation. +$$ +\mathcal{G}(\boldsymbol{X}^{(i)} + \mathrm{d}\boldsymbol{X}^{(i)}) \approx \mathcal{G}(\boldsymbol{X}^{(i)}) + \left.\frac{\partial \mathcal{G}}{\partial \boldsymbol{X}}\right|_{\boldsymbol{X}^{(i)}} \cdot \mathrm{d}\boldsymbol{X}^{(i)} = 0 +$$ +$$ +\mathrm{d}\boldsymbol{X}^{(i)} = - \left.\frac{\partial \mathcal{G}}{\partial \boldsymbol{X}}\right|_{\boldsymbol{X}^{(i)}}^{-1} \cdot \mathcal{G}(\boldsymbol{X}^{(i)}) +\tag{2.1} +$$ +With $\frac{\partial \mathcal{G}}{\partial \boldsymbol{X}}$ being the Jacobian of the function $\mathcal{G}$ : +$$ +\frac{\partial \mathcal{G}}{\partial \boldsymbol{X}} = +\begin{pmatrix} +\frac{\partial \mathcal{G}_0}{\partial \boldsymbol{X}_0} & \frac{\partial \mathcal{G}_0}{\partial \boldsymbol{X}_1} & \dots & \frac{\partial \mathcal{G}_0}{\partial \boldsymbol{X}_n} \\ +\frac{\partial \mathcal{G}_1}{\partial \boldsymbol{X}_0} & \frac{\partial \mathcal{G}_1}{\partial \boldsymbol{X}_1} & \dots & \frac{\partial \mathcal{G}_1}{\partial \boldsymbol{X}_n} \\ +\vdots & \vdots & \ddots & \vdots \\ +\frac{\partial \mathcal{G}_n}{\partial \boldsymbol{X}_0} & \frac{\partial \mathcal{G}_n}{\partial \boldsymbol{X}_1} & \dots & \frac{\partial \mathcal{G}_n}{\partial \boldsymbol{X}_n} +\end{pmatrix} +$$ +Then the current guess is updated according to +$$ +\boldsymbol{X}^{(i+1)} = \boldsymbol{X}^{(i)} + \mathrm{d}\boldsymbol{X}^{(i)} +\tag{2.2} +$$ +The algorithm stops when the norm of the current residue $\|\boldsymbol{r}^{(i)}\| = \|\mathcal{G}(\boldsymbol{X}^{(i)})\|$ is under a certain threshold $\epsilon \in \mathbb{R}$ -### Build the linear matrix system +#### 2.2) Newton-Raphson for dynamic equations of motion +##### 2.2.1) Acceleration-based integration scheme +In the case of dynamic equations of motion, we have defined the function for which we want to find the root (1.4). Now we are going to write the recurrence relations resulting for the application of a Newton-Raphson algorithm to this problem. -The left hand side matrix $\mathbf{A}$ is built using the function: -``` cpp -matrix = MechanicalMatrix(r_M, r_B, r_K); -``` -where $r_M$ (mass coefficient), $r_B$ (damping coefficient) and $r_K$ (stiffness coefficient) are Rayleigh coefficients (see section below). Depending on the scheme (explicit or implicit, see previous paragraph) and on the type of LinearSolver used (if any), the abstract function `MechanicalMatrix` will trigger different [visitors](../../visitors/), thus different functions to compute the system matrix $\mathbf{A}$. Discover the API used for the computation of $\mathbf{A}$ in the [ForceField](../../multi-model-representation/forcefield/#forcefield-api) and [Mass](../../multi-model-representation/mass/#mass-api) doc pages. +By applying (2.1) to (1.4) we can compute the final linear system to solve: +$$ +\left.\frac{\partial \mathcal{G}_{t}}{\partial \boldsymbol{X}}\right|_{\boldsymbol{X}^{(i)}} \mathrm{d}\boldsymbol{X}^{(i)} = - \mathcal{G}_{t}(\boldsymbol{X}^{(i)}) +$$ -The right hand side vector *b* is built through the function: -``` cpp -computeForce(b) -``` +Let's take a look closer to the Jacobian. -Again, Depending on the scheme (explicit or implicit, see previous paragraph), the abstract function `computeForce` will trigger different [visitors](../../visitors/), thus different functions to accumulate the forces into the vector $b$. Discover the API used for the computation of $b$ in the [ForceField](../../multi-model-representation/forcefield/#forcefield-api) doc page. +$$ +\begin{aligned} +\left.\frac{\partial \mathcal{G}_{t}}{\partial \boldsymbol{X}}\right|_{\boldsymbol{X}^{(i)}} &= +\begin{pmatrix} +\left.\frac{\partial \mathcal{G}_{t,0}}{\partial \boldsymbol{X}_{\boldsymbol{x}}}\right|_{\boldsymbol{X}^{(i)}} & \left.\frac{\partial \mathcal{G}_{t,0}}{\partial \boldsymbol{X}_{\boldsymbol{v}}}\right|_{\boldsymbol{X}^{(i)}} & \left.\frac{\partial \mathcal{G}_{t,0}}{\partial \boldsymbol{X}_{\boldsymbol{a}}}\right|_{\boldsymbol{X}^{(i)}} \\ +\left.\frac{\partial \mathcal{G}_{t,1}}{\partial \boldsymbol{X}_{\boldsymbol{x}}}\right|_{\boldsymbol{X}^{(i)}} & \left.\frac{\partial \mathcal{G}_{t,1}}{\partial \boldsymbol{X}_{\boldsymbol{v}}}\right|_{\boldsymbol{X}^{(i)}} & \left.\frac{\partial \mathcal{G}_{t,1}}{\partial \boldsymbol{X}_{\boldsymbol{a}}}\right|_{\boldsymbol{X}^{(i)}} \\ +\left.\frac{\partial \mathcal{G}_{t,2}}{\partial \boldsymbol{X}_{\boldsymbol{x}}}\right|_{\boldsymbol{X}^{(i)}} & \left.\frac{\partial \mathcal{G}_{t,2}}{\partial \boldsymbol{X}_{\boldsymbol{v}}}\right|_{\boldsymbol{X}^{(i)}} & \left.\frac{\partial \mathcal{G}_{t,2}}{\partial \boldsymbol{X}_{\boldsymbol{a}}}\right|_{\boldsymbol{X}^{(i)}} +\end{pmatrix} \\ +&= +\begin{pmatrix} +-\left.\frac{\partial \mathcal{F}}{\partial \boldsymbol{x}}\right|_{\boldsymbol{X}^{(i)}} & \quad - \left.\frac{\partial \mathcal{F}}{\partial \boldsymbol{v}}\right|_{\boldsymbol{X}^{(i)}} & \quad \boldsymbol{M} \\ +I & \quad -\left.\frac{\mathrm{d} g_{\boldsymbol{x}}^{(t,h)}}{\mathrm{d} \boldsymbol{v}}\right|_{\boldsymbol{X}^{(i)}} & \quad -\left.\frac{\mathrm{d} g_{\boldsymbol{x}}^{(t,h)}}{\mathrm{d} \boldsymbol{a}}\right|_{\boldsymbol{X}^{(i)}} \\ +0 & \quad I & \quad -\left.\frac{\mathrm{d} g_{\boldsymbol{v}}^{(t,h)}}{\mathrm{d} \boldsymbol{a}}\right|_{\boldsymbol{X}^{(i)}} +\end{pmatrix} +\end{aligned} +$$ +We can already identify two common mechanical matrices : +$$ +\begin{cases} +\boldsymbol{K}^{(i)} = \left.\frac{\partial \mathcal{F}}{\partial \boldsymbol{x}}\right|_{\boldsymbol{X}^{(i)}} \\ +\boldsymbol{B}^{(i)} = \left.\frac{\partial \mathcal{F}}{\partial \boldsymbol{v}}\right|_{\boldsymbol{X}^{(i)}} +\end{cases} +$$ +We are going to inverse this linear system by bloc by injecting the last equation into the second one. This is done by first inverting the last equation with respect to $\mathrm{d}\boldsymbol{X}^{(i)}_{\boldsymbol{v}}$. +$$ +\mathrm{d}\boldsymbol{X}^{(i)}_{\boldsymbol{v}} = \left.\frac{\mathrm{d} g_{\boldsymbol{v}}^{(t,h)}}{\mathrm{d} \boldsymbol{a}}\right|_{\boldsymbol{X}^{(i)}} \mathrm{d}\boldsymbol{X}^{(i)}_{\boldsymbol{a}} - \mathcal{G}_{t}(\boldsymbol{X}^{(i)})_2 +$$ +Then we can inject it back to the second equation and then invert it again with respect to $\mathrm{d}\boldsymbol{X}^{(i)}_{\boldsymbol{x}}$ +$$ +\begin{aligned} +\mathrm{d}\boldsymbol{X}^{(i)}_{\boldsymbol{x}} &= \left.\frac{\mathrm{d} g_{\boldsymbol{x}}^{(t,h)}}{\mathrm{d} \boldsymbol{v}}\right|_{\boldsymbol{X}^{(i)}} \left(\left.\frac{\mathrm{d} g_{\boldsymbol{v}}^{(t,h)}}{\mathrm{d} \boldsymbol{a}}\right|_{\boldsymbol{X}^{(i)}} \mathrm{d}\boldsymbol{X}^{(i)}_{\boldsymbol{a}} - \mathcal{G}_{t}(\boldsymbol{X}^{(i)})_2\right) + \left.\frac{\mathrm{d} g_{\boldsymbol{x}}^{(t,h)}}{\mathrm{d} \boldsymbol{a}}\right|_{\boldsymbol{X}^{(i)}} \mathrm{d}\boldsymbol{X}^{(i)}_{\boldsymbol{a}} - \mathcal{G}_{t}(\boldsymbol{X}^{(i)})_1 \\ +&= \left(\left.\frac{\mathrm{d} g_{\boldsymbol{x}}^{(t,h)}}{\mathrm{d} \boldsymbol{v}}\right|_{\boldsymbol{X}^{(i)}}\left.\frac{\mathrm{d} g_{\boldsymbol{v}}^{(t,h)}}{\mathrm{d} \boldsymbol{a}}\right|_{\boldsymbol{X}^{(i)}} + \left.\frac{\mathrm{d} g_{\boldsymbol{x}}^{(t,h)}}{\mathrm{d} \boldsymbol{a}}\right|_{\boldsymbol{X}^{(i)}}\right) \mathrm{d}\boldsymbol{X}^{(i)}_{\boldsymbol{a}} - \left.\frac{\mathrm{d} g_{\boldsymbol{x}}^{(t,h)}}{\mathrm{d} \boldsymbol{v}}\right|_{\boldsymbol{X}^{(i)}}\mathcal{G}_{t}(\boldsymbol{X}^{(i)})_2 - \mathcal{G}_{t}(\boldsymbol{X}^{(i)})_1 +\end{aligned} +$$ -### State vectors in IntegrationScheme -In order to build the linear matrix system, the IntegrationScheme uses information contained in [state vectors](../../mechanicalobject/#state-vectors) (like DOFs and their derivatives) within the scope of the IntegrationScheme. The IntegrationScheme does not access the state vectors directly. It accesses the state vectors remotely using visitors, which traverse the graph starting from the node which contains the solver. This keeps the implementation of the solver independent of the simulated objects and their types. +Let define the following coefficients +$$ +\begin{aligned} +\text{DG}^{(i)}_{\boldsymbol{v}} &= \left.\frac{\mathrm{d} g_{\boldsymbol{v}}^{(t,h)}}{\mathrm{d} \boldsymbol{a}}\right|_{\boldsymbol{X}^{(i)}} \\ +\text{DG}^{(i)}_{\boldsymbol{x}} &= \left(\left.\frac{\mathrm{d} g_{\boldsymbol{x}}^{(t,h)}}{\mathrm{d} \boldsymbol{v}}\right|_{\boldsymbol{X}^{(i)}} \left.\frac{\mathrm{d} g_{\boldsymbol{v}}^{(t,h)}}{\mathrm{d} \boldsymbol{a}}\right|_{\boldsymbol{X}^{(i)}} + \left.\frac{\mathrm{d} g_{\boldsymbol{x}}^{(t,h)}}{\mathrm{d} \boldsymbol{a}}\right|_{\boldsymbol{X}^{(i)}}\right) +\end{aligned} +$$ +We can inject the two expressions of $\mathrm{d}\boldsymbol{X}^{(i)}_{\boldsymbol{x}}$ and $\mathrm{d}\boldsymbol{X}^{(i)}_{\boldsymbol{v}}$ into the first equation : +$$ +\begin{aligned} +&-\boldsymbol{K}^{(i)}\mathrm{d}\boldsymbol{X}^{(i)}_{\boldsymbol{x}} - \boldsymbol{B}^{(i)}\mathrm{d}\boldsymbol{X}^{(i)}_{\boldsymbol{v}} + \boldsymbol{M} \mathrm{d}\boldsymbol{X}^{(i)}_{\boldsymbol{a}} = -\mathcal{G}_{t}(\boldsymbol{X}^{(i)})_0 \\ +\Longleftrightarrow\quad& -\boldsymbol{K}^{(i)} \left(\text{DG}^{(i)}_{\boldsymbol{x}} \mathrm{d}\boldsymbol{X}^{(i)}_{\boldsymbol{a}} - \left.\frac{\mathrm{d} g_{\boldsymbol{x}}^{(t,h)}}{\mathrm{d} \boldsymbol{v}}\right|_{\boldsymbol{X}^{(i)}}\mathcal{G}_{t}(\boldsymbol{X}^{(i)})_2 - \mathcal{G}_{t}(\boldsymbol{X}^{(i)})_1\right) - \boldsymbol{B}^{(i)} \left(\text{DG}^{(i)}_{\boldsymbol{v}} \mathrm{d}\boldsymbol{X}^{(i)}_{\boldsymbol{a}} - \mathcal{G}_{t}(\boldsymbol{X}^{(i)})_2\right) + \boldsymbol{M} \mathrm{d}\boldsymbol{X}^{(i)}_{\boldsymbol{a}} = -\mathcal{G}_{t}(\boldsymbol{X}^{(i)})_0 \\ +\Longleftrightarrow\quad& (\boldsymbol{M} -\boldsymbol{K}^{(i)}\cdot\text{DG}^{(i)}_{\boldsymbol{x}} - \boldsymbol{B}^{(i)} \text{DG}^{(i)}_{\boldsymbol{v}})\mathrm{d}\boldsymbol{X}^{(i)}_{\boldsymbol{a}} = -\mathcal{G}_{t}(\boldsymbol{X}^{(i)})_0-\boldsymbol{K}^{(i)} \mathcal{G}_{t}(\boldsymbol{X}^{(i)})_1 - \left(\boldsymbol{B}^{(i)} + \boldsymbol{K}^{(i)}\left.\frac{\mathrm{d} g_{\boldsymbol{x}}^{(t,h)}}{\mathrm{d} \boldsymbol{v}}\right|_{\boldsymbol{X}^{(i)}}\right) \mathcal{G}_{t}(\boldsymbol{X}^{(i)})_2 +\end{aligned} +$$ -Each type of solver may use different auxiliary state vectors to implement their simulation method. State vectors (MultiVec) are allocated and processed in the scope of the solver in a thread-safe way using an instance of _simulation::common::VectorOperations_. For instance, a Runge-Kutta algorithms needs to save the result of previous time steps. +To summarize, the update strategy is as follow : +$$ +\begin{cases} +\mathrm{d}\boldsymbol{X}^{(i)}_{\boldsymbol{a}} &= (\boldsymbol{M} -\boldsymbol{K}^{(i)}\cdot\text{DG}^{(i)}_{\boldsymbol{x}} - \boldsymbol{B}^{(i)} \text{DG}^{(i)}_{\boldsymbol{v}})^{(-1)} \left(-\mathcal{G}_{t}(\boldsymbol{X}^{(i)})_0-\boldsymbol{K}^{(i)} \mathcal{G}_{t}(\boldsymbol{X}^{(i)})_1 - \left(\boldsymbol{B}^{(i)} + \boldsymbol{K}^{(i)}\left.\frac{\mathrm{d} g_{\boldsymbol{x}}^{(t,h)}}{\mathrm{d} \boldsymbol{v}}\right|_{\boldsymbol{X}^{(i)}}\right)\cdot \mathcal{G}_{t}(\boldsymbol{X}^{(i)})_2\right) \\ +\mathrm{d}\boldsymbol{X}^{(i)}_{\boldsymbol{v}} &= \text{DG}^{(i)}_{\boldsymbol{v}} \mathrm{d}\boldsymbol{X}^{(i)}_{\boldsymbol{a}} - \mathcal{G}_{t}(\boldsymbol{X}^{(i)})_2 \\ +\mathrm{d}\boldsymbol{X}^{(i)}_{\boldsymbol{x}} &= \left.\frac{\mathrm{d} g_{\boldsymbol{x}}^{(t,h)}}{\mathrm{d} \boldsymbol{v}}\right|_{\boldsymbol{X}^{(i)}} \mathrm{d}\boldsymbol{X}^{(i)}_{\boldsymbol{v}} + \left.\frac{\mathrm{d} g_{\boldsymbol{x}}^{(t,h)}}{\mathrm{d} \boldsymbol{a}}\right|_{\boldsymbol{X}^{(i)}} \mathrm{d}\boldsymbol{X}^{(i)}_{\boldsymbol{a}} - \mathcal{G}_{t}(\boldsymbol{X}^{(i)})_1 \\ +&= \text{DG}^{(i)}_{\boldsymbol{x}} \mathrm{d}\boldsymbol{X}^{(i)}_{\boldsymbol{a}} - \left.\frac{\mathrm{d} g_{\boldsymbol{x}}^{(t,h)}}{\mathrm{d} \boldsymbol{v}}\right|_{\boldsymbol{X}^{(i)}}\mathcal{G}_{t}(\boldsymbol{X}^{(i)})_2 - \mathcal{G}_{t}(\boldsymbol{X}^{(i)})_1 \\ +\boldsymbol{X}^{(i+1)} &= \boldsymbol{X}^{(i)} + \begin{pmatrix} \mathrm{d}\boldsymbol{X}^{(i)}_{\boldsymbol{x}} \\ \mathrm{d}\boldsymbol{X}^{(i)}_{\boldsymbol{v}} \\ \mathrm{d}\boldsymbol{X}^{(i)}_{\boldsymbol{a}} \end{pmatrix} +\end{cases} +\tag{2.3} +$$ -To create an auxiliary vector, this can be done as follows: -``` cpp -MultiVecCoord pos(&vop, core::VecCoordId::position() ); // standard position vector -MultiVecDeriv acc(&vop); // auxiliary vector -MultiVecCoord previousPos(&vop, previousPosID); // additional vector -``` +##### 2.2.1) Velocity-based integration scheme +In the case of velocity-based equations of motion, we have defined the function for which we want to find the root (1.6). Now we are going to write the recurrence relations resulting for the application of a Newton-Raphson algorithm to this problem. +By applying (2.1) to (1.6) we can compute the final linear system to solve: +$$ +\left.\frac{\partial \mathcal{G}_{t}}{\partial \tilde{\boldsymbol{X}}}\right|_{\tilde{\boldsymbol{X}}^{(i)}} \mathrm{d}\tilde{\boldsymbol{X}}^{(i)} = - \mathcal{G}_{t}(\tilde{\boldsymbol{X}}^{(i)}) +$$ +Let's take a look closer to the Jacobian. +$$ +\begin{aligned} +\left.\frac{\partial \mathcal{G}_{t}}{\partial \tilde{\boldsymbol{X}}}\right|_{\tilde{\boldsymbol{X}}^{(i)}} &= +\begin{pmatrix} +\left.\frac{\partial \mathcal{G}_{t,0}}{\partial \tilde{\boldsymbol{X}}_{\boldsymbol{x}}}\right|_{\tilde{\boldsymbol{X}}^{(i)}} & \left.\frac{\partial \mathcal{G}_{t,0}}{\partial \tilde{\boldsymbol{X}}_{\boldsymbol{v}}}\right|_{\tilde{\boldsymbol{X}}^{(i)}} \\ +\left.\frac{\partial \mathcal{G}_{t,1}}{\partial \tilde{\boldsymbol{X}}_{\boldsymbol{x}}}\right|_{\tilde{\boldsymbol{X}}^{(i)}} & \left.\frac{\partial \mathcal{G}_{t,1}}{\partial \tilde{\boldsymbol{X}}_{\boldsymbol{v}}}\right|_{\tilde{\boldsymbol{X}}^{(i)}} +\end{pmatrix} \\ +&= +\begin{pmatrix} +-\left.\frac{\partial \mathcal{F}}{\partial \boldsymbol{x}}\right|_{\tilde{\boldsymbol{X}}^{(i)}} & \quad \boldsymbol{M} \left.\frac{\mathrm{d} g_{\boldsymbol{v}}^{(t,h)-1}}{\mathrm{d} \boldsymbol{v}}\right|_{\tilde{\boldsymbol{X}}^{(i)}} - \left.\frac{\partial \mathcal{F}}{\partial \boldsymbol{v}}\right|_{\tilde{\boldsymbol{X}}^{(i)}} \\ +I & \quad -\left.\frac{\mathrm{d} \tilde{g}_{\boldsymbol{x}}^{(t,h)}}{\mathrm{d} \boldsymbol{v}}\right|_{\tilde{\boldsymbol{X}}^{(i)}} +\end{pmatrix} +\end{aligned} +$$ +We can already identify two common mechanical matrices : +$$ +\begin{cases} +\boldsymbol{K}^{(i)} = \left.\frac{\partial \mathcal{F}}{\partial \boldsymbol{x}}\right|_{\tilde{\boldsymbol{X}}^{(i)}} \\ +\boldsymbol{B}^{(i)} = \left.\frac{\partial \mathcal{F}}{\partial \boldsymbol{v}}\right|_{\tilde{\boldsymbol{X}}^{(i)}} +\end{cases} +$$ +This linear system by bloc can be solved using a Shur complement method. Here we have a special case where all blocks are squared. So we can do a development using the bottom left block: +$$ +\mathrm{d}\tilde{\boldsymbol{X}}^{(i)}_{\boldsymbol{x}} = \left.\frac{\mathrm{d} \tilde{g}_{\boldsymbol{x}}^{(t,h)}}{\mathrm{d} \boldsymbol{v}}\right|_{\tilde{\boldsymbol{X}}^{(i)}} \cdot \mathrm{d}\tilde{\boldsymbol{X}}^{(i)}_{\boldsymbol{v}} - \mathcal{G}_{t}(\tilde{\boldsymbol{X}}^{(i)})_1 +$$ +And by injecting this into the first equation we get : +$$ +\begin{aligned} +& -\boldsymbol{K}^{(i)} \left(\left.\frac{\mathrm{d} \tilde{g}_{\boldsymbol{x}}^{(t,h)}}{\mathrm{d} \boldsymbol{v}}\right|_{\tilde{\boldsymbol{X}}^{(i)}} \cdot \mathrm{d}\tilde{\boldsymbol{X}}^{(i)}_{\boldsymbol{v}} - \mathcal{G}_{t}(\tilde{\boldsymbol{X}}^{(i)})_1\right) + \left(\boldsymbol{M} \left.\frac{\mathrm{d} g_{\boldsymbol{v}}^{(t,h)-1}}{\mathrm{d} \boldsymbol{v}}\right|_{\tilde{\boldsymbol{X}}^{(i)}} - \boldsymbol{B}^{(i)}\right)\mathrm{d}\tilde{\boldsymbol{X}}^{(i)}_{\boldsymbol{v}} = - \mathcal{G}_{t}(\tilde{\boldsymbol{X}}^{(i)})_0 \\ +\Longleftrightarrow & \left( \boldsymbol{M} \left.\frac{\mathrm{d} g_{\boldsymbol{v}}^{(t,h)-1}}{\mathrm{d} \boldsymbol{v}}\right|_{\tilde{\boldsymbol{X}}^{(i)}} - \boldsymbol{K}^{(i)} \left.\frac{\mathrm{d} \tilde{g}_{\boldsymbol{x}}^{(t,h)}}{\mathrm{d} \boldsymbol{v}}\right|_{\tilde{\boldsymbol{X}}^{(i)}} - \boldsymbol{B}^{(i)}\right) \mathrm{d}\tilde{\boldsymbol{X}}^{(i)}_{\boldsymbol{v}} = -\boldsymbol{K}^{(i)} \mathcal{G}_{t}(\tilde{\boldsymbol{X}}^{(i)})_1 - \mathcal{G}_{t}(\tilde{\boldsymbol{X}}^{(i)})_0 +\end{aligned} +\tag{2.4} +$$ +To summarize, the update strategy is as follow : +$$ +\begin{cases} +\mathrm{d}\tilde{\boldsymbol{X}}^{(i)}_{\boldsymbol{v}} &= \left( \boldsymbol{M} \left.\frac{\mathrm{d} g_{\boldsymbol{v}}^{(t,h)-1}}{\mathrm{d} \boldsymbol{v}}\right|_{\tilde{\boldsymbol{X}}^{(i)}} - \boldsymbol{K}^{(i)} \left.\frac{\mathrm{d} \tilde{g}_{\boldsymbol{x}}^{(t,h)}}{\mathrm{d} \boldsymbol{v}}\right|_{\tilde{\boldsymbol{X}}^{(i)}} - \boldsymbol{B}^{(i)}\right)^{-1} \left(-\boldsymbol{K}^{(i)} \mathcal{G}_{t}(\tilde{\boldsymbol{X}}^{(i)})_1 - \mathcal{G}_{t}(\tilde{\boldsymbol{X}}^{(i)})_0\right) \\ +\mathrm{d}\tilde{\boldsymbol{X}}^{(i)}_{\boldsymbol{x}} &= \left.\frac{\mathrm{d} \tilde{g}_{\boldsymbol{x}}^{(t,h)}}{\mathrm{d} \boldsymbol{v}}\right|_{\tilde{\boldsymbol{X}}^{(i)}} \cdot \mathrm{d}\tilde{\boldsymbol{X}}^{(i)}_{\boldsymbol{v}} - \mathcal{G}_{t}(\tilde{\boldsymbol{X}}^{(i)})_1 \\ +\tilde{\boldsymbol{X}}^{(i+1)} &= \tilde{\boldsymbol{X}}^{(i)} + \begin{pmatrix} \mathrm{d}\tilde{\boldsymbol{X}}^{(i)}_{\boldsymbol{x}} \\ \mathrm{d}\tilde{\boldsymbol{X}}^{(i)}_{\boldsymbol{v}} \end{pmatrix} +\end{cases} +\tag{2.5} +$$ +In the SOFA code +---------------- -### Compute the solution +### Design choices +In the light of the presented Newton-Raphson algorithm and the associated equations, the design for implicit intergation scheme is now clearer : it needs an API that enable the computation of such iterative algorithm, meaning that we need to be able compute the right-hand-side of the linearized equation independently from the left-hand-side, and also update the solution independently. This choice has lead to the following API interface: -In most cases, the matrix system $\mathbf{A}x=b$ can then be sent to a [LinearSolver](../../system-resolution/linear-solver/) in charge of finally solving the system defined according to the chosen scheme. Within the function *IntegrationScheme::solve()*, the call to the LinearSolver will appear through the function call: +#### ImplicitIntegrationScheme +This base class inherits directly from the class `BaseIntegrationScheme` and proposes the virtual API taht'll be implemented by the different implicit integrations chemes. Here is the list of thoses methods : -``` cpp -matrix.solve(x, b); -``` +```cpp -Some simple matrix cases provides a diagonal matrix $\mathbf{A}$. In this specific configuration, a solution can directly be found by dividing the right hand side vector *b* by the diagonal matrix $\mathbf{A}$. This is done using the function: -``` cpp -mop.accFromF(acc, f); ``` +#### AccelerationBasedIntegrationScheme -Rayleigh damping ----------------- - -The Rayleigh damping is a numerical damping. This damping has therefore no physical meaning and must not be mixed up with physical damping (like _DiagonalVelocityDampingForceField_ in SOFA). The Rayleigh damping corresponds to a damping matrix that is proportional to the mass or/and stiffness matrices using coefficients, respectively Rayleigh stiffness factor $r_K$ or Rayleigh mass factor $r_M$. This numerical damping is usually used to stabilize or ease convergence of the simulation. However, it has to be used carefully. +#### VelocityBasedIntegrationScheme -When Rayleigh damping is used, the damping matrix becomes the sum of the physical and the numerical (Rayleigh) damping: $\mathbf{B} = \mathbf{B}_{\text{phys}} - \mathbf{M} \cdot r_M+ \mathbf{K} \cdot r_K$ where $\mathbf{B}_{\text{phys}}$ is the physical damping matrix, $\mathbf{M}$ is the mass matrix and $\mathbf{K}$ is the stiffness matrix. -The negative sign in front of $\mathbf{M}$, a positive matrix, represents the fact that viscosity opposes motion. Elasticity also opposes it, however $\mathbf{K}$ is a negative matrix. This formula therefore provides two positive coefficients $r_K$ and $r_M$. -You can see the use of Rayleigh mass and stiffness dampings in the `solve()` function of the _EulerImplicit_ class (see EulerImplicitSolver.cpp). +#### Special case : StaticEquilibriumIntegrationscheme +> **Note on Rayleigh damping :**\ +> Most of the integration scheme propose to add _Rayleigh damping_ which is a numerical damping. This damping has therefore no physical meaning and must not be mixed up with physical damping (like _DiagonalVelocityDampingForceField_ in SOFA). The Rayleigh damping corresponds to a damping matrix that is proportional to the mass or/and stiffness matrices using coefficients, respectively Rayleigh stiffness factor $r_K$ or Rayleigh mass factor $r_M$. This numerical damping is usually used to stabilize or ease convergence of the simulation. However, it has to be used carefully. +> +> When Rayleigh damping is used, the damping matrix becomes the sum of the physical and the numerical (Rayleigh) damping: $\mathbf{B} = \mathbf{B}_{\text{phys}} - \mathbf{M} \cdot r_M+ \mathbf{K} \cdot r_K$ where $\mathbf{B}_{\text{phys}}$ is the physical damping matrix, $\mathbf{M}$ is the mass matrix and $\mathbf{K}$ is the stiffness matrix. +> The negative sign in front of $\mathbf{M}$, a positive matrix, represents the fact that viscosity opposes motion. Elasticity also opposes it, however $\mathbf{K}$ is a negative matrix. This formula therefore provides two positive coefficients $r_K$ and $r_M$. +> +> You can see the use of Rayleigh mass and stiffness dampings in the `integrate()` function of the _EulerImplicit_ class (see EulerImplicitSolver.cpp). From 53d44841019859872dabbb5ac7eb78c5efcbae80 Mon Sep 17 00:00:00 2001 From: Paul Baksic Date: Thu, 30 Jul 2026 17:29:17 +0200 Subject: [PATCH 04/10] Finalize the main principles --- .../10_Integration_Scheme.md | 111 +++++++++++++++++- 1 file changed, 108 insertions(+), 3 deletions(-) diff --git a/20_Simulation_Principles/40_System_Resolution/10_Integration_Scheme.md b/20_Simulation_Principles/40_System_Resolution/10_Integration_Scheme.md index 6edab3a99..37bcc9a0a 100644 --- a/20_Simulation_Principles/40_System_Resolution/10_Integration_Scheme.md +++ b/20_Simulation_Principles/40_System_Resolution/10_Integration_Scheme.md @@ -323,16 +323,52 @@ In the light of the presented Newton-Raphson algorithm and the associated equati This base class inherits directly from the class `BaseIntegrationScheme` and proposes the virtual API taht'll be implemented by the different implicit integrations chemes. Here is the list of thoses methods : ```cpp + //This method purpose is to enable the integration scheme to prepare internal state before beginning the solving step. (Not pure virtual) + virtual void doSetupIntegrationStep(...) {} + // Compute the system matrix. The boolean i shere to avoid computin zero-valued vectors after the first iteration (i.e. such as the integration scheme error which is always null after one iteration) + virtual void computeLHS(bool firstIteration = false) = 0; + + // compute the current RHS. + virtual void computeRHS(bool firstIteration = false) = 0; + + // Returns the evaluation of the residual + virtual SReal evaluateResidual() = 0; + + // Solve the linear equation from a Newton iteration, i.e. it computes (x^{i+1}-x^i). + virtual void solveLinearEquation() = 0; + + // Use the computed unknown to update state accordingly. The alpha parameter is here to implement line search + virtual void updateStatesFromLinearSolution(SReal alpha, bool firstIteration = false) = 0; + + //This method is called after the integration step is completed. (Not pure virutal) + virtual void finalizeIntegrationStep() {} + + // This method comes form the BaseIntegrationScheme API, it is a monolithic step integration. + virtual void integrate(...) override; + + // This methods returns the factor to put in front of the linear system unknown accumulating it to the velocity. + // In the case of acceleration-based integraiton scheme, we can see in (2.3) that this should return $DG_v$ + virtual SReal getVelocityIntegrationFactor() const = 0; + + // This methods returns the factor to put in front of the linear system unknown accumulating it to the position. + // In the case of acceleration-based integraiton scheme, we can see in (2.3) that this should return $DG_x$ + virtual SReal getPositionIntegrationFactor() const = 0; + + // This method returns the order of the integration scheme in term of number of past timestep needed to compute the next timestep. For instance, if $p_{t+dt} = f(v_{t+dt}, ... , v_{t-k*dt}$, then the order is k+1 + virtual sofa::Size getIntegrationSchemeTimeOrder() const = 0; + + // Rayleigh damping coefficient related to stiffness > 0 + Data d_rayleighStiffness; / + // Rayleigh damping coefficient related to mass > 0 + Data d_rayleighMass; ``` -#### AccelerationBasedIntegrationScheme +For the two famillies a lot of this API can be implemented agnostically from the integraiton s cheme expression. Knowing the integration scheme expression is finally only required to compute the residual vector, and some factors of the LHS or RHS computation. Knowing this, we have proposed two specialization of this class, proposing new simplier API entries for both acceleration and velocity based integration scheme. -#### VelocityBasedIntegrationScheme -#### Special case : StaticEquilibriumIntegrationscheme > **Note on Rayleigh damping :**\ > Most of the integration scheme propose to add _Rayleigh damping_ which is a numerical damping. This damping has therefore no physical meaning and must not be mixed up with physical damping (like _DiagonalVelocityDampingForceField_ in SOFA). The Rayleigh damping corresponds to a damping matrix that is proportional to the mass or/and stiffness matrices using coefficients, respectively Rayleigh stiffness factor $r_K$ or Rayleigh mass factor $r_M$. This numerical damping is usually used to stabilize or ease convergence of the simulation. However, it has to be used carefully. > @@ -340,3 +376,72 @@ This base class inherits directly from the class `BaseIntegrationScheme` and pro > The negative sign in front of $\mathbf{M}$, a positive matrix, represents the fact that viscosity opposes motion. Elasticity also opposes it, however $\mathbf{K}$ is a negative matrix. This formula therefore provides two positive coefficients $r_K$ and $r_M$. > > You can see the use of Rayleigh mass and stiffness dampings in the `integrate()` function of the _EulerImplicit_ class (see EulerImplicitSolver.cpp). + + +#### AccelerationBasedIntegrationScheme + +For acceleration-based integration scheme, the only parts that are integration dependent are : + +- $\text{DG}^{(i)}_{\boldsymbol{x}} = \left.\frac{\mathrm{d} g_{\boldsymbol{v}}^{(t,h)}}{\mathrm{d} \boldsymbol{a}}\right|_{\boldsymbol{X}^{(i)}} $ +- $\text{DG}^{(i)}_{\boldsymbol{v}} = \left(\left.\frac{\mathrm{d} g_{\boldsymbol{x}}^{(t,h)}}{\mathrm{d} \boldsymbol{v}}\right|_{\boldsymbol{X}^{(i)}} \left.\frac{\mathrm{d} g_{\boldsymbol{v}}^{(t,h)}}{\mathrm{d} \boldsymbol{a}}\right|_{\boldsymbol{X}^{(i)}} + \left.\frac{\mathrm{d} g_{\boldsymbol{x}}^{(t,h)}}{\mathrm{d} \boldsymbol{a}}\right|_{\boldsymbol{X}^{(i)}}\right)$ +- $\mathcal{G}_{t}(\boldsymbol{X}^{(i)})_1$ +- $\mathcal{G}_{t}(\boldsymbol{X}^{(i)})_2$ + +This enable to completely fulfill the `ImplicitIntegrationScheme` API in this new class by adding new light methods that will need to be specialized for every integration scheme : +```cpp +// This method returns a scalar which is the value of the derivative of the position integration scheme with respect to the acceleration. +// To build the $DG_x$ and $DG_v$ factor (see equation above) +virtual SReal getPositionUpdateDerivedFromAcceleration() const = 0; + +// This method returns a scalar which is the value of the derivative of the position integration scheme with respect to the velocity. +// To build the $DG_x$ and $DG_v$ factor (see equation above) +virtual SReal getPositionUpdateDerivedFromVelocity() const = 0; + +// This method returns a scalar which is the value of the derivative of the velocity integration scheme with respect to the acceleration. +// To build the $DG_x$ and $DG_v$ factor (see equation above) +virtual SReal getVelocityUpdateDerivedFromAcceleration() const = 0; + +// This method compute the error in term of position update given the current state, or $G_t(X)_1$ +virtual void computeCurrentPositionIntegrationError(...) = 0; + +// This method compute the error in term of velocity update given the current state, or $G_t(X)_2$ +virtual void computeCurrentVelocityIntegrationError(...) = 0; +``` + +The tree first method returning only scalar values, they are the most traightforward method to implement. The two last have to deal with advanced concept of SOFA such as mechanical operation on `VecId`. For an example on how to implement this, see the Newmak implementation [here](//TODO, link to cpp file in the master branch once the PR is merged). + + +#### VelocityBasedIntegrationScheme + + +For velocity-based integration scheme, the only parts that are integration dependent are : + +- $g_{\boldsymbol{v}}^{(t,h)-1}$ +- $\left.\frac{\mathrm{d} \tilde{g}_{\boldsymbol{x}}^{(t,h)}}{\mathrm{d} \boldsymbol{v}}\right|_{\tilde{\boldsymbol{X}}^{(i)}}$ +- $\left.\frac{\mathrm{d} g_{\boldsymbol{v}}^{(t,h)-1}}{\mathrm{d} \boldsymbol{v}}\right|_{\tilde{\boldsymbol{X}}^{(i)}}$ +- $\mathcal{G}_{t}(\boldsymbol{X}^{(i)})_1$ + +This enable to completely fulfill the `ImplicitIntegrationScheme` API in this new class by adding new light methods that will need to be specialized for every integration scheme : + +```cpp +// This method returns a scalar which is the value of the derivative of the position integration scheme with respect to the velocity. +// To update the state +virtual SReal getPositionUpdateDerivedFromVelocity() const = 0; + +// This method returns a scalar which is the value of the derivative of the position integration scheme with respect to the velocity. +// To build the RHS +virtual SReal getInverseVelocityUpdateDerivedFromVelocity() const = 0; + +//This method compute the error in term of position update, or $G_t(X)_1$ +virtual void computeCurrentPositionIntegrationError(...) = 0; + +//This method compute the acceleration given the current velocity, or $g_v^{(t,h)-1}$ +virtual void computeAccelerationFromVelocity(...) = 0; +``` +Again, the two first method returning only scalar values, they are the most traightforward method to implement. The two last have to deal with advanced concept of SOFA such as mechanical operation on `VecId`. For an example on how to implement this, see the Euler implicit implementation [here](//TODO, link to cpp file in the master branch once the PR is merged). + +#### Special case : StaticEquilibriumIntegrationscheme + +The Static equilibrium integration scheme is a special case as it is not a real integraiton scheme becaus eit does not advance time linearly. + +For more details see its dedicated [documentation page](../../../components/integrationscheme/backward/staticequilibriumintegrationscheme/) From 952d7fff8b9518a14bb3c1b965c391a844850c9b Mon Sep 17 00:00:00 2001 From: Paul Baksic Date: Thu, 30 Jul 2026 17:32:33 +0200 Subject: [PATCH 05/10] Rename folder/files --- .../10_EulerExplicitIntegrationScheme.md} | 0 .../20_EulerImplicitIntegrationScheme.md} | 0 .../50_StaticEquilibriumIntegrationScheme.md} | 0 .../60_NewmarkImplicitIntegrationScheme.md} | 0 .../20_Backward/BDFIntegrationScheme.md} | 0 .../20_Backward/NewtonRaphsonSolver.md | 47 ------------------- 6 files changed, 47 deletions(-) rename 30_Components/{40_ODESolver/10_Forward/10_EulerExplicitSolver.md => 40_IntegrationScheme/10_Forward/10_EulerExplicitIntegrationScheme.md} (100%) rename 30_Components/{40_ODESolver/20_Backward/20_EulerImplicitSolver.md => 40_IntegrationScheme/20_Backward/20_EulerImplicitIntegrationScheme.md} (100%) rename 30_Components/{40_ODESolver/20_Backward/50_StaticSolver.md => 40_IntegrationScheme/20_Backward/50_StaticEquilibriumIntegrationScheme.md} (100%) rename 30_Components/{40_ODESolver/20_Backward/60_NewmarkImplicitSolver.md => 40_IntegrationScheme/20_Backward/60_NewmarkImplicitIntegrationScheme.md} (100%) rename 30_Components/{40_ODESolver/20_Backward/BDFOdeSolver.md => 40_IntegrationScheme/20_Backward/BDFIntegrationScheme.md} (100%) delete mode 100644 30_Components/40_ODESolver/20_Backward/NewtonRaphsonSolver.md diff --git a/30_Components/40_ODESolver/10_Forward/10_EulerExplicitSolver.md b/30_Components/40_IntegrationScheme/10_Forward/10_EulerExplicitIntegrationScheme.md similarity index 100% rename from 30_Components/40_ODESolver/10_Forward/10_EulerExplicitSolver.md rename to 30_Components/40_IntegrationScheme/10_Forward/10_EulerExplicitIntegrationScheme.md diff --git a/30_Components/40_ODESolver/20_Backward/20_EulerImplicitSolver.md b/30_Components/40_IntegrationScheme/20_Backward/20_EulerImplicitIntegrationScheme.md similarity index 100% rename from 30_Components/40_ODESolver/20_Backward/20_EulerImplicitSolver.md rename to 30_Components/40_IntegrationScheme/20_Backward/20_EulerImplicitIntegrationScheme.md diff --git a/30_Components/40_ODESolver/20_Backward/50_StaticSolver.md b/30_Components/40_IntegrationScheme/20_Backward/50_StaticEquilibriumIntegrationScheme.md similarity index 100% rename from 30_Components/40_ODESolver/20_Backward/50_StaticSolver.md rename to 30_Components/40_IntegrationScheme/20_Backward/50_StaticEquilibriumIntegrationScheme.md diff --git a/30_Components/40_ODESolver/20_Backward/60_NewmarkImplicitSolver.md b/30_Components/40_IntegrationScheme/20_Backward/60_NewmarkImplicitIntegrationScheme.md similarity index 100% rename from 30_Components/40_ODESolver/20_Backward/60_NewmarkImplicitSolver.md rename to 30_Components/40_IntegrationScheme/20_Backward/60_NewmarkImplicitIntegrationScheme.md diff --git a/30_Components/40_ODESolver/20_Backward/BDFOdeSolver.md b/30_Components/40_IntegrationScheme/20_Backward/BDFIntegrationScheme.md similarity index 100% rename from 30_Components/40_ODESolver/20_Backward/BDFOdeSolver.md rename to 30_Components/40_IntegrationScheme/20_Backward/BDFIntegrationScheme.md diff --git a/30_Components/40_ODESolver/20_Backward/NewtonRaphsonSolver.md b/30_Components/40_ODESolver/20_Backward/NewtonRaphsonSolver.md deleted file mode 100644 index 442b1e1e0..000000000 --- a/30_Components/40_ODESolver/20_Backward/NewtonRaphsonSolver.md +++ /dev/null @@ -1,47 +0,0 @@ -NewtonRaphsonSolver -=================== - -NewtonRaphsonSolver is a component able to solve nonlinear equations using [Newton-Raphson method](https://en.wikipedia.org/wiki/Newton%27s_method). -From an initial guess, the algorithm successively computes better approximations of the root of the nonlinear function. -At every iteration, multiple criteria are evaluated to decide to stop the algorithm (because it converged or the maximum number of iterations has been reached), or to continue. - -The algorithm relies on the derivative of the function and a linear system to solve. -For a function $F : \mathbb{R}^k \rightarrow \mathbb{R}^k$, the new approximation of the root $x_{n+1}$ is computed as: - -$$ -\nabla_F(x_n) (x_{n+1} - x_n) = -F(x_n) -$$ - -where: - -- $x_i$ is the $i$-th approximation of the root -- $x_0$ is the initial guess -- $\nabla_F$ is the Jacobian matrix of the function - -If $dx$ is the solution of the previous linear system, then - -$$ -x_{n+1} = dx + x_n -$$ - -Example -------- - -To solve a static equilibrium (see [StaticSolver](StaticSolver.md)), the nonlinear equation to solve is the sum of forces must be equal to zero ($\sum F = 0$). At each iteration, the linear system $K dx = -\sum F$ must be solved to compute the next approximation of the root. Here K is the derivative of the forces, also called the stiffness matrix. - -Usage ------ - -This component must be linked by another component requiring to solve a nonlinear equation, such as an implicit ODE solver or a static solver. - -In XML format, the link may look like: - -```xml - - -``` From 07cecc3d1e47591093f792f6e9b853d62e9f082f Mon Sep 17 00:00:00 2001 From: Paul Baksic Date: Thu, 30 Jul 2026 17:38:19 +0200 Subject: [PATCH 06/10] Minimal renaming in IS files --- .../10_Forward/10_EulerExplicitIntegrationScheme.md | 12 ++++++------ .../20_Backward/20_EulerImplicitIntegrationScheme.md | 10 +++++----- .../50_StaticEquilibriumIntegrationScheme.md | 12 ++++++------ .../60_NewmarkImplicitIntegrationScheme.md | 10 +++++----- .../20_Backward/BDFIntegrationScheme.md | 4 ++-- 5 files changed, 24 insertions(+), 24 deletions(-) diff --git a/30_Components/40_IntegrationScheme/10_Forward/10_EulerExplicitIntegrationScheme.md b/30_Components/40_IntegrationScheme/10_Forward/10_EulerExplicitIntegrationScheme.md index 8b8152dd4..2196a6746 100644 --- a/30_Components/40_IntegrationScheme/10_Forward/10_EulerExplicitIntegrationScheme.md +++ b/30_Components/40_IntegrationScheme/10_Forward/10_EulerExplicitIntegrationScheme.md @@ -1,7 +1,7 @@ -EulerExplicitSolver +EulerExplicitIntegrationScheme =================== -The EulerExplicitSolver component belongs to the category of [integration schemes or ODE Solver](../../../../simulation-principles/system-resolution/integration-scheme/). This scheme allows to solve dynamic systems explicitly: all forces will be computed based on the state information at the current time step $x(t)$. +The EulerExplicitIntegrationScheme component belongs to the category of [integration schemes](../../../../simulation-principles/system-resolution/integration-scheme/). This scheme allows to solve dynamic systems explicitly: all forces will be computed based on the state information at the current time step $x(t)$. Looking at continuum mechanics, the linear system $\mathbf{A}x=b$ arises from the dynamic equation. This dynamic is written as follows but other physics (like heat transfer) result in a similar equation: @@ -9,7 +9,7 @@ $$ \mathbf{M}\Delta v=dt\left(f(x,t)\right) $$ -where $x$ is the degrees of freedom, $\mathbf{M}$ the mass matrix and $f(x,t)$ a function of $x$ (and possibly its derivatives) acting on our system. In the case of the EulerExplicitSolver, this equation can be written: +where $x$ is the degrees of freedom, $\mathbf{M}$ the mass matrix and $f(x,t)$ a function of $x$ (and possibly its derivatives) acting on our system. In the case of the EulerExplicitIntegrationScheme, this equation can be written: $$ \mathbf{M}\Delta v=dt\left(f(x(t))\right) @@ -26,16 +26,16 @@ Depending on whether the mass matrix is diagonal or not, SOFA supports two cases Note that the **symplectic** data allows to modify the scheme to make it [symplectic](https://en.wikipedia.org/wiki/Semi-implicit_Euler_method), i.e. velocities are updated before the positions. It allows to update the positions from the newly computed velocities, instead of velocities from the previous time step. This option makes the scheme more robust in time. -EulerExplicitSolver is symplectic by default. +EulerExplicitIntegrationScheme is symplectic by default. Sequence diagram ---------------- - + Usage ----- -The EulerExplicitSolver **requires** a MechanicalObject to store the state vectors. However, as explained above, no LinearSolver is needed and the EulerExplicitSolver is **only working using a [UniformMass](../../../mass/uniformmass/) or [DiagonalMass](../../../mass/diagonalmass/)**, which ensures to have a diagonal system matrix. +The EulerExplicitIntegrationScheme **requires** a MechanicalObject to store the state vectors. However, as explained above, no LinearSolver is needed and the EulerExplicitIntegrationScheme is **only working using a [UniformMass](../../../mass/uniformmass/) or [DiagonalMass](../../../mass/diagonalmass/)**, which ensures to have a diagonal system matrix. diff --git a/30_Components/40_IntegrationScheme/20_Backward/20_EulerImplicitIntegrationScheme.md b/30_Components/40_IntegrationScheme/20_Backward/20_EulerImplicitIntegrationScheme.md index 3c9e85035..cc861ec1f 100644 --- a/30_Components/40_IntegrationScheme/20_Backward/20_EulerImplicitIntegrationScheme.md +++ b/30_Components/40_IntegrationScheme/20_Backward/20_EulerImplicitIntegrationScheme.md @@ -1,7 +1,7 @@ -EulerImplicitSolver +EulerImplicitIntegrationScheme =================== -This component belongs to the category of [integration schemes or ODE Solver](../../../../simulation-principles/system-resolution/integration-scheme/). This scheme builds the system following an implicit scheme: forces are considered based on the state information at the next time step $x(t+dt)$, unknown at the current time step. +This component belongs to the category of [integration schemes](../../../../simulation-principles/system-resolution/integration-scheme/). This scheme builds the system following an implicit scheme: forces are considered based on the state information at the next time step $x(t+dt)$, unknown at the current time step. Looking at continuum mechanics, the linear system $\mathbf{A}x=b$ arises from the dynamic equation. This dynamic is written as follows but other physics (like heat transfer) result in a similar equation: @@ -9,7 +9,7 @@ $$ \mathbf{M}\Delta v=dt\left(f(x,t)\right) $$ -where $x$ is the degrees of freedom, $\mathbf{M}$ the mass matrix and $f(x,t)$ a function of $x$ (and possibly its derivatives) acting on our system. In the case of the EulerImplicitSolver, this equation can be written: +where $x$ is the degrees of freedom, $\mathbf{M}$ the mass matrix and $f(x,t)$ a function of $x$ (and possibly its derivatives) acting on our system. In the case of the EulerImplicitIntegrationScheme, this equation can be written: $$ \mathbf{M} \Delta v=dt \cdot f(x(t+dt)) @@ -102,7 +102,7 @@ $$ Sequence diagram ---------------- - + @@ -110,7 +110,7 @@ Sequence diagram Usage ----- -The EulerImplicitSolver **requires**: +The EulerImplicitIntegrationScheme **requires**: - a [LinearSolver](../../../../simulation-principles/system-resolution/linear-solver/) to solve the linear system - and a MechanicalObject to store the state vectors. diff --git a/30_Components/40_IntegrationScheme/20_Backward/50_StaticEquilibriumIntegrationScheme.md b/30_Components/40_IntegrationScheme/20_Backward/50_StaticEquilibriumIntegrationScheme.md index 270be8cac..41f05f4f1 100644 --- a/30_Components/40_IntegrationScheme/20_Backward/50_StaticEquilibriumIntegrationScheme.md +++ b/30_Components/40_IntegrationScheme/20_Backward/50_StaticEquilibriumIntegrationScheme.md @@ -1,7 +1,7 @@ -StaticSolver +StaticEquilibriumIntegrationScheme ============ -This component belongs to the category of [integration schemes or ODE Solver](../../../../simulation-principles/system-resolution/integration-scheme/). +This component belongs to the category of [integration schemes](../../../../simulation-principles/system-resolution/integration-scheme/). In the field of mechanics, statics consists in finding the equilibrium taking into account the loads (internal forces, external forces and torques) acting on the physical system, that do not experience an acceleration ( $a=0$ ). Finding a static equilibrium means finding a solution to: $\textstyle \sum F=0$ where $F$ is the sum of all loads, one of which might be unknown. @@ -9,7 +9,7 @@ In a static analysis, the inertia and damping effects are ignored, i.e. the dyna In a static simulation involving elasticity, the linear system that we solve corresponds to $K \Delta u=f$ where $K$ is the stiffness matrix (derivative of elastic forces), $\Delta u$ is a vector describing the total increment of displacement and $f$ are all explicit forces. We realize here that the static solver is in fact an implicit scheme, since the $K$ matrix is present in the left-hand side of the equation. The solution $\Delta u$ is obtained iteratively. At each iteration _i_, the displacement is incremented $\Delta u_{i+1}=\Delta u_{i}+\delta u_i$, thus resulting in the following system to solve: $K_i \delta u_i=f$. -In case of non-linear elasticity, $K_i$ is a linearization which must be updated with regards to the increment of displacement $\delta u_i$. In such cases, several iterations of Newton Raphson are required to find an appropriate approximate solution. In one step of the StaticSolver, the number of Newton Raphson iterations is ruled by the data field **newton_iterations**. +In case of non-linear elasticity, $K_i$ is a linearization which must be updated with regards to the increment of displacement $\delta u_i$. In such cases, several iterations of Newton Raphson are required to find an appropriate approximate solution. In one step of the StaticEquilibriumIntegrationScheme, the number of Newton Raphson iterations is ruled by the data field **newton_iterations**. _Reminder_: the [Newton Raphson method](https://en.wikipedia.org/wiki/Newton%27s_method) is an iterative algorithm aiming at finding the solution of the system $f(x)=0$ where $f(x)$ is non-linear. At each iteration of Newton Raphson algorithm, we find a new approximate solution: @@ -25,17 +25,17 @@ $$ Sequence diagram ---------------- - + Usage ----- -At each simulation step and each Newton Raphson iteration, the StaticSolver **requires**: +At each simulation step and each Newton Raphson iteration, the StaticEquilibriumIntegrationScheme **requires**: - a [LinearSolver](../../../../simulation-principles/system-resolution/linear-solver/) to solve the linear system - and a MechanicalObject to store the state vectors. -A StaticSolver must be used in simulations where the dynamics has no or a negligible effect on the system. A StaticSolver would also be relevant for systems with low mass. In such case, we fall into the quasi-static analysis. +A StaticEquilibriumIntegrationScheme must be used in simulations where the dynamics has no or a negligible effect on the system. A StaticEquilibriumIntegrationScheme would also be relevant for systems with low mass. In such case, we fall into the quasi-static analysis. In some loading configuration, applying the full forces and torques might not lead to any converging simulation. It is then relevant to go for an incremental loading, i.e. loads are applied incrementally at each simulation step $i$. This incremental loading has to be done in the associated ForceField. If you want to use this solver with Newton Raphson iterations, it is in the user's hand to make sure the external forces used in the scene (pressure, traction, etc.) only get incremented at each time step, and not at each calls to addForce (which is currently the case for most force fields). diff --git a/30_Components/40_IntegrationScheme/20_Backward/60_NewmarkImplicitIntegrationScheme.md b/30_Components/40_IntegrationScheme/20_Backward/60_NewmarkImplicitIntegrationScheme.md index 6c9a84d4a..bf60b8311 100644 --- a/30_Components/40_IntegrationScheme/20_Backward/60_NewmarkImplicitIntegrationScheme.md +++ b/30_Components/40_IntegrationScheme/20_Backward/60_NewmarkImplicitIntegrationScheme.md @@ -1,9 +1,9 @@ -NewmarkImplicitSolver +NewmarkIntegrationScheme ===================== -This component belongs to the category of [integration schemes or ODE Solver](../../../../simulation-principles/system-resolution/integration-scheme/). +This component belongs to the category of [integration schemes](../../../../simulation-principles/system-resolution/integration-scheme/). -This scheme is an implicit time integrator for dynamic system using the Newmark scheme. To compute the new position or new velocity, the NewmarkImplicitSolver is based on the following equations: +This scheme is an implicit time integrator for dynamic system using the Newmark scheme. To compute the new position or new velocity, the NewmarkIntegrationScheme is based on the following equations: $$ x_{t+h}=x_t+h v_t+\frac{h^2}{2}((1-2\beta)a_t+2\beta a_{t+h}) @@ -42,13 +42,13 @@ $$ Sequence diagram ---------------- - + Usage ----- -At each simulation step and each Newton Raphson iteration, the NewmarkImplicitSolver **requires**: +At each simulation step and each Newton Raphson iteration, the NewmarkIntegrationScheme **requires**: - a [LinearSolver](../../../../simulation-principles/system-resolution/linear-solver/) to solve the linear system - and a MechanicalObject to store the state vectors. diff --git a/30_Components/40_IntegrationScheme/20_Backward/BDFIntegrationScheme.md b/30_Components/40_IntegrationScheme/20_Backward/BDFIntegrationScheme.md index 51d1b82a7..2a95fbddf 100644 --- a/30_Components/40_IntegrationScheme/20_Backward/BDFIntegrationScheme.md +++ b/30_Components/40_IntegrationScheme/20_Backward/BDFIntegrationScheme.md @@ -1,7 +1,7 @@ -BDFOdeSolver +BDFIntegrationScheme ============ -This component belongs to the category of [integration schemes or ODE Solver](../../../../simulation-principles/system-resolution/integration-scheme/). +This component belongs to the category of [integration schemes](../../../../simulation-principles/system-resolution/integration-scheme/). It is an implicit method for the numerical integration of the ODE resulting from Newton's second law of motion. The method relies on [Backward Differentiation Formula](https://en.wikipedia.org/wiki/Backward_differentiation_formula) (BDF). From d858bf22c971488a4bdb9ce059d235112cd7c5ac Mon Sep 17 00:00:00 2001 From: Paul Baksic Date: Thu, 30 Jul 2026 17:42:27 +0200 Subject: [PATCH 07/10] Remove flowcharts as they are really quickly outdated --- .../integrationscheme/EulerExplicitSolver.png | Bin 27540 -> 0 bytes .../integrationscheme/EulerImplicitSolver.png | Bin 128953 -> 0 bytes .../NewmarkImplicitSolver.png | Bin 55942 -> 0 bytes images/integrationscheme/StaticSolver.png | Bin 117365 -> 0 bytes 4 files changed, 0 insertions(+), 0 deletions(-) delete mode 100644 images/integrationscheme/EulerExplicitSolver.png delete mode 100644 images/integrationscheme/EulerImplicitSolver.png delete mode 100644 images/integrationscheme/NewmarkImplicitSolver.png delete mode 100644 images/integrationscheme/StaticSolver.png diff --git a/images/integrationscheme/EulerExplicitSolver.png b/images/integrationscheme/EulerExplicitSolver.png deleted file mode 100644 index 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From 458f8c17809002f118c19b036fefa5f0ee8ffc64 Mon Sep 17 00:00:00 2001 From: Paul Baksic Date: Fri, 31 Jul 2026 10:47:55 +0200 Subject: [PATCH 08/10] Add some insights on the API --- .../20_MechanicalObject.md | 2 +- .../10_Integration_Scheme.md | 26 ++++++++++++++----- 2 files changed, 21 insertions(+), 7 deletions(-) diff --git a/20_Simulation_Principles/20_MechanicalObject.md b/20_Simulation_Principles/20_MechanicalObject.md index a019770d1..a7007d796 100644 --- a/20_Simulation_Principles/20_MechanicalObject.md +++ b/20_Simulation_Principles/20_MechanicalObject.md @@ -122,7 +122,7 @@ Symbolic ids The MultiVec entries are not directly accessible by the solvers. The MultiVec are represented by identificators. The operations on the vectors are implemented using visitors which contain the identificators of the relevant vectors. The MultiVec identificators (MultiVecId) have different types, depending on the data they contain (positions or their derivatives) and the access mode. -The use of symbolic identificators (MultiVecId) prevent other components (like solvers) from handling state vectors directly and allow to easily work with abstract MultiVec by using their ids. These symbolic ids are widely used by specialized visitors, like the ones used in [ODESolver](./../system-resolution/integration-scheme/). +The use of symbolic identificators (MultiVecId) prevent other components (like solvers) from handling state vectors directly and allow to easily work with abstract MultiVec by using their ids. These symbolic ids are widely used by specialized visitors, like the ones used in [IntegrationScheme](./../system-resolution/integration-scheme/). ``` cpp typedef TMultiVecId ConstMultiVecCoordId; diff --git a/20_Simulation_Principles/40_System_Resolution/10_Integration_Scheme.md b/20_Simulation_Principles/40_System_Resolution/10_Integration_Scheme.md index 37bcc9a0a..354fea650 100644 --- a/20_Simulation_Principles/40_System_Resolution/10_Integration_Scheme.md +++ b/20_Simulation_Principles/40_System_Resolution/10_Integration_Scheme.md @@ -313,7 +313,7 @@ $$ $$ -In the SOFA code +The SOFA implementation ---------------- ### Design choices @@ -348,11 +348,11 @@ This base class inherits directly from the class `BaseIntegrationScheme` and pro virtual void integrate(...) override; // This methods returns the factor to put in front of the linear system unknown accumulating it to the velocity. - // In the case of acceleration-based integraiton scheme, we can see in (2.3) that this should return $DG_v$ + // In the case of acceleration-based integration scheme, we can see in (2.3) that this should return $DG_v$ virtual SReal getVelocityIntegrationFactor() const = 0; // This methods returns the factor to put in front of the linear system unknown accumulating it to the position. - // In the case of acceleration-based integraiton scheme, we can see in (2.3) that this should return $DG_x$ + // In the case of acceleration-based integration scheme, we can see in (2.3) that this should return $DG_x$ virtual SReal getPositionIntegrationFactor() const = 0; // This method returns the order of the integration scheme in term of number of past timestep needed to compute the next timestep. For instance, if $p_{t+dt} = f(v_{t+dt}, ... , v_{t-k*dt}$, then the order is k+1 @@ -365,7 +365,7 @@ This base class inherits directly from the class `BaseIntegrationScheme` and pro ``` -For the two famillies a lot of this API can be implemented agnostically from the integraiton s cheme expression. Knowing the integration scheme expression is finally only required to compute the residual vector, and some factors of the LHS or RHS computation. Knowing this, we have proposed two specialization of this class, proposing new simplier API entries for both acceleration and velocity based integration scheme. +For the two famillies a lot of this API can be implemented agnostically from the integration s cheme expression. Knowing the integration scheme expression is finally only required to compute the residual vector, and some factors of the LHS or RHS computation. Knowing this, we have proposed two specialization of this class, proposing new simplier API entries for both acceleration and velocity based integration scheme. @@ -440,8 +440,22 @@ virtual void computeAccelerationFromVelocity(...) = 0; ``` Again, the two first method returning only scalar values, they are the most traightforward method to implement. The two last have to deal with advanced concept of SOFA such as mechanical operation on `VecId`. For an example on how to implement this, see the Euler implicit implementation [here](//TODO, link to cpp file in the master branch once the PR is merged). -#### Special case : StaticEquilibriumIntegrationscheme +#### Special case : StaticEquilibriumIntegrationScheme -The Static equilibrium integration scheme is a special case as it is not a real integraiton scheme becaus eit does not advance time linearly. +The Static equilibrium integration scheme is a special case as it is not a real integration scheme becaus eit does not advance time linearly. For more details see its dedicated [documentation page](../../../components/integrationscheme/backward/staticequilibriumintegrationscheme/) + +#### Some API details + +For all integration scheme, as we have seen in the dynamic equations, the right-hand-side and the left-hand-side are composed of contirbutions coming from the mass and external and internal forces. This means that for building those terms, the intergation schemes will need to access the ForceFields and the Mass objects to either accumulate their _explicit_ part in the right-hand-side or their _implicit_ part in the left-hand-side. + +This is performed using a mechanism of visitors that will iterate through the scene graph and call API method of those component so that they can add their contributions. For instance : +- `addForce` method of forcefield will contribute to the right-hand-side +- `buildStiffnessMatrix` method of forcefield will build its implicit part, contributing to the left-hand-side +- `addMDx` in mass object will be used to add the gravity term in the right-hand-side + +This list is not exhausitve, please refer to the [Forcefield](../../multi-model-representation/forcefield/) and [Mass](../../multi-model-representation/mass/) documentation pages. + + + From ddf4efef8602aba918f5da33b04e63361703f417 Mon Sep 17 00:00:00 2001 From: Paul Baksic Date: Fri, 31 Jul 2026 11:49:40 +0200 Subject: [PATCH 09/10] Add components rewritting --- .../10_Integration_Scheme.md | 4 + .../10_EulerExplicitIntegrationScheme.md | 6 - .../20_EulerImplicitIntegrationScheme.md | 104 +++++------------- .../50_StaticEquilibriumIntegrationScheme.md | 8 +- .../60_NewmarkImplicitIntegrationScheme.md | 51 ++++----- .../20_Backward/BDFIntegrationScheme.md | 83 +++++++------- 6 files changed, 93 insertions(+), 163 deletions(-) diff --git a/20_Simulation_Principles/40_System_Resolution/10_Integration_Scheme.md b/20_Simulation_Principles/40_System_Resolution/10_Integration_Scheme.md index 354fea650..65a7d8afc 100644 --- a/20_Simulation_Principles/40_System_Resolution/10_Integration_Scheme.md +++ b/20_Simulation_Principles/40_System_Resolution/10_Integration_Scheme.md @@ -440,6 +440,10 @@ virtual void computeAccelerationFromVelocity(...) = 0; ``` Again, the two first method returning only scalar values, they are the most traightforward method to implement. The two last have to deal with advanced concept of SOFA such as mechanical operation on `VecId`. For an example on how to implement this, see the Euler implicit implementation [here](//TODO, link to cpp file in the master branch once the PR is merged). +> **Note :**\ +> The velocity-base integration schemes offer a possibility to reduce the integration to a _first order_ integration, meaning the velocity is considered as null at the begining of each time step. This feature can help for quasi-static simulation or simulations where the objects dynamic is by nature subject to numerical noise such as very lightwheight objects. \ +> This feature can be activated using the data `firstOrder=True` + #### Special case : StaticEquilibriumIntegrationScheme The Static equilibrium integration scheme is a special case as it is not a real integration scheme becaus eit does not advance time linearly. diff --git a/30_Components/40_IntegrationScheme/10_Forward/10_EulerExplicitIntegrationScheme.md b/30_Components/40_IntegrationScheme/10_Forward/10_EulerExplicitIntegrationScheme.md index 2196a6746..b160047e8 100644 --- a/30_Components/40_IntegrationScheme/10_Forward/10_EulerExplicitIntegrationScheme.md +++ b/30_Components/40_IntegrationScheme/10_Forward/10_EulerExplicitIntegrationScheme.md @@ -28,12 +28,6 @@ It allows to update the positions from the newly computed velocities, instead of This option makes the scheme more robust in time. EulerExplicitIntegrationScheme is symplectic by default. -Sequence diagram ----------------- - - - - Usage ----- diff --git a/30_Components/40_IntegrationScheme/20_Backward/20_EulerImplicitIntegrationScheme.md b/30_Components/40_IntegrationScheme/20_Backward/20_EulerImplicitIntegrationScheme.md index cc861ec1f..00a12edfd 100644 --- a/30_Components/40_IntegrationScheme/20_Backward/20_EulerImplicitIntegrationScheme.md +++ b/30_Components/40_IntegrationScheme/20_Backward/20_EulerImplicitIntegrationScheme.md @@ -3,106 +3,56 @@ EulerImplicitIntegrationScheme This component belongs to the category of [integration schemes](../../../../simulation-principles/system-resolution/integration-scheme/). This scheme builds the system following an implicit scheme: forces are considered based on the state information at the next time step $x(t+dt)$, unknown at the current time step. -Looking at continuum mechanics, the linear system $\mathbf{A}x=b$ arises from the dynamic equation. This dynamic is written as follows but other physics (like heat transfer) result in a similar equation: +This is the most broadly used integration scheme thank's to its unconditional stability and simplicity. -$$ -\mathbf{M}\Delta v=dt\left(f(x,t)\right) -$$ - -where $x$ is the degrees of freedom, $\mathbf{M}$ the mass matrix and $f(x,t)$ a function of $x$ (and possibly its derivatives) acting on our system. In the case of the EulerImplicitIntegrationScheme, this equation can be written: - -$$ -\mathbf{M} \Delta v=dt \cdot f(x(t+dt)) -$$ +It is recommended to read the theoretical part of the [integration schemes](../../../../simulation-principles/system-resolution/integration-scheme/) page to understand the following. -by using a Taylor expansion, we get: +The EulerImplicitIntegrationScheme inherits from VelocityBaseIntegrationScheme because its integration scheme in velocity is invertible. Its equations are the following : $$ -\mathbf{M} \Delta v=dt \cdot \left( f(x(t))+\cdot \frac{\partial f}{\partial x} \Delta x \right) +\begin{aligned} +g_{\boldsymbol{x}}^{(t,h)} &: \boldsymbol{v}, \boldsymbol{a} \mapsto \boldsymbol{x}_t + h \boldsymbol{v} \\ +g_{\boldsymbol{v}}^{(t,h)} &: \boldsymbol{a} \mapsto \boldsymbol{v}_t + h \boldsymbol{a} +\end{aligned} $$ -since we have: $\Delta x=dt(v(t)+\Delta v)$, then: - -$$ -\mathbf{M}\Delta v=dt\cdot \left( f(x(t))+dt\cdot \frac{\partial f}{\partial x}v(t)+dt\cdot \frac{\partial f}{\partial x} \Delta v \right) -$$ - -Finally, gathering the unknown (depending on $\Delta v$) in the left-hand side, we have: - -$$ -\left( \mathbf{M}-dt^2 \cdot \frac{\partial f}{\partial x} \right) \Delta v=dt\cdot f(x(t))+dt^2\cdot \frac{\partial f}{\partial x}v(t) -$$ - -We can notice the appearance of the stiffness matrix : $\mathbf{K}_{ij}=\textstyle\frac{\partial f_i}{\partial x_j}$. The stiffness matrix $\mathbf{K}$ is a symmetric matrix, can either be linear or non-linear regarding $x$. - -$$ -\left( \mathbf{M}-dt^2 \cdot \mathbf{K} \right) \Delta v=dt\cdot f(x(t))+dt^2\cdot \mathbf{K}v(t) -$$ - -The computation of the **right hand side** is done by the ForceFields. Just like in the explicit case (see [EulerExplicitSolver](../../forward/eulerexplicitsolver/)), the explicit contribution $dt\left(f(x(t))\right)$ is implemented in the same function `addForce()`. The second part $dt^2\cdot \frac{\partial f}{\partial x}v(t)$ is computed by the function `addDForce()`. - -It is important to note that, depending on the **choice of LinearSolver** (direct or iterative), the API functions called to build the **left hand side** system matrix $\mathbf{A}=\left( M-dt^2 \cdot \frac{\partial f}{\partial x} \right)$ will not be the same: - - - if a direct solver is used, the mass $\mathbf{M}$ is computed in the `addMToMatrix()` and the stiffness part $-dt^2 \cdot \frac{\partial f}{\partial x}$ is computed in the function `addKToMatrix()` in ForceFields - - - if an iterative solver is used, the mass is iteratively multiplied by the unknown $\mathbf{M} \Delta v$ within the `addMDx()`, as the stiffness part $-dt^2 \cdot \frac{\partial f}{\partial x} \Delta v$ within the function `addDForce()` in ForceFields. +#### API specialization -#### Considering viscosity - - -As you might have noticed, the Taylor expansion detailed above does not take into account a possible dependency of the force $f(x,t)$ on the velocity. By considering it, the effect of velocity will result in a viscosity effect through the damping matrix $\mathbf{B}$. - -Let's apply the Taylor expansion taking into account the velocity and we get: - -$$ -\mathbf{M} \Delta v=dt \cdot \left( f(x(t), v(t))+\cdot \frac{\partial f}{\partial x} \Delta x+\cdot \frac{\partial f}{\partial v} \Delta v \right) +As explained in the [IntegrationScheme](../../../../simulation-principles/system-resolution/integration-scheme/) documentation, the specilization consist in implementing 4 methods that requires the knowledge of four terms/expressions. for Euler implicit they are the following : $$ - +\begin{aligned} +&\tilde{g}_{\boldsymbol{x}}^{(t,h)} \equiv g_{\boldsymbol{x}}^{(t,h)} \\ +&g_{\boldsymbol{v}}^{(t,h)-1} : \boldsymbol{v} \mapsto \frac{1}{h}(\boldsymbol{v} - \boldsymbol{v}_t) +\end{aligned} +\qquad \qquad \qquad +\begin{aligned} +&\frac{\mathrm{d} \tilde{g}_{\boldsymbol{x}}^{(t,h)}}{\mathrm{d} \boldsymbol{v}} = h \\ +&\frac{\mathrm{d} g_{\boldsymbol{v}}^{(t,h)-1}}{\mathrm{d} \boldsymbol{v}} = \frac{1}{h} +\end{aligned} $$ -\left( \mathbf{M}-dt \cdot \frac{\partial f}{\partial v}-dt^2 \cdot \frac{\partial f}{\partial x} \right) \Delta v=dt\cdot f(x(t),v(t))+dt^2\cdot \frac{\partial f}{\partial x}v(t) -$$ - -$$ -\left( \mathbf{M}-dt \cdot \mathbf{B}-dt^2 \cdot \mathbf{K} \right) \Delta v=dt\cdot f(x(t),v(t))+dt^2\cdot \mathbf{K}v(t) -$$ - -Depending on the choice of LinearSolver (direct or iterative), the API functions called to build the $\mathbf{B}$ damping matrix on the left hand side will not be the same: - - - if a direct solver is used, the damping matrix $\mathbf{B}$ is computed in the `addBToMatrix()` in ForceFields - - if an iterative solver is used, the damping is iteratively multiplied by the unknown $\mathbf{B} \Delta v$ within the `addDForce()` just as the stiffness part in the function `addDForce()` in ForceFields. - - - -#### Dissipation - -SOFA is a framework aiming at interactive simulations. For this purpose, dissipative schemes are very appropriate. The Euler scheme is an order 1 time integration scheme (since only using the current state $x(t)$ and no older one like $x(t-dt)$). It is known to be a dissipative scheme. Moreover, only one Newton step is performed in the EulerImplicit, which might harm the energy conservation. - -#### Numerical damping - -With Rayleigh damping, the option is given to the user to add numerical damping. The description of the meaning and effect of these Rayleigh damping coefficients is given in [ODESolver](../../../../simulation-principles/system-resolution/integration-scheme/#rayleigh-damping). #### Trapezoidal rule -Activating the trapezoidalScheme option of the Euler implicit scheme will make the scheme less dissipative. This is due to the fact that the [trapezoidal rule](https://en.wikipedia.org/wiki/Trapezoidal_rule) increases the order of the time integration. Moreover, higher order schemes are known to be less dissipative. +Activating the trapezoidalScheme option of the Euler implicit scheme will make the scheme less dissipative. This will apply the [trapezoidal rule](https://en.wikipedia.org/wiki/Trapezoidal_rule) to the velocity integration, and thus will increase the order of the time integration, which is known to be less dissipative. It is also known to increase robustness and stability to the time integration due to the order 2 in time of this trapezoidal scheme. The modified scheme is the following: $$ -y_{n+1}-y_n=\frac{dt}{2}(f(y_{n+1})+f(y_n)) +\begin{aligned} +g_{\boldsymbol{x}}^{(t,h)} &: \boldsymbol{v}, \boldsymbol{a} \mapsto \boldsymbol{x}_t + \frac{h}{2} \boldsymbol{v_t} + \frac{h}{2} \boldsymbol{v} \\ +g_{\boldsymbol{v}}^{(t,h)} &: \boldsymbol{a} \mapsto \boldsymbol{v}_t + h \boldsymbol{a} +\end{aligned} $$ -This results in the following linear system: +This results in the updated term: $$ -\left( \mathbf{M}-\frac{dt^2}{4} \frac{\partial f}{\partial x}\right) \Delta v=dt\cdot f(x(t))+\frac{dt^2}{2}\cdot \frac{\partial f}{\partial x}v(t) -$$ +\frac{\mathrm{d} \tilde{g}_{\boldsymbol{x}}^{(t,h)}}{\mathrm{d} \boldsymbol{v}} = h/2 +$$ -Sequence diagram ----------------- - - +To activate this trapezoidal rule, you need to use the data `trapezoidalScheme=true`. diff --git a/30_Components/40_IntegrationScheme/20_Backward/50_StaticEquilibriumIntegrationScheme.md b/30_Components/40_IntegrationScheme/20_Backward/50_StaticEquilibriumIntegrationScheme.md index 41f05f4f1..08a805175 100644 --- a/30_Components/40_IntegrationScheme/20_Backward/50_StaticEquilibriumIntegrationScheme.md +++ b/30_Components/40_IntegrationScheme/20_Backward/50_StaticEquilibriumIntegrationScheme.md @@ -11,7 +11,7 @@ In a static simulation involving elasticity, the linear system that we solve cor In case of non-linear elasticity, $K_i$ is a linearization which must be updated with regards to the increment of displacement $\delta u_i$. In such cases, several iterations of Newton Raphson are required to find an appropriate approximate solution. In one step of the StaticEquilibriumIntegrationScheme, the number of Newton Raphson iterations is ruled by the data field **newton_iterations**. -_Reminder_: the [Newton Raphson method](https://en.wikipedia.org/wiki/Newton%27s_method) is an iterative algorithm aiming at finding the solution of the system $f(x)=0$ where $f(x)$ is non-linear. At each iteration of Newton Raphson algorithm, we find a new approximate solution: +_Reminder_: the Newton Raphson method is an iterative algorithm aiming at finding the solution of the system $f(x)=0$ where $f(x)$ is non-linear. At each iteration of Newton Raphson algorithm, we find a new approximate solution: $x^{n+1}=x^n-\frac{f(x^n)}{f'(x^n)}$ where $f'(x^n) = \frac{df}{dx}(x^n)$ @@ -22,12 +22,6 @@ $$ $$ -Sequence diagram ----------------- - - - - Usage ----- diff --git a/30_Components/40_IntegrationScheme/20_Backward/60_NewmarkImplicitIntegrationScheme.md b/30_Components/40_IntegrationScheme/20_Backward/60_NewmarkImplicitIntegrationScheme.md index bf60b8311..5c617d5fd 100644 --- a/30_Components/40_IntegrationScheme/20_Backward/60_NewmarkImplicitIntegrationScheme.md +++ b/30_Components/40_IntegrationScheme/20_Backward/60_NewmarkImplicitIntegrationScheme.md @@ -1,48 +1,37 @@ NewmarkIntegrationScheme ===================== -This component belongs to the category of [integration schemes](../../../../simulation-principles/system-resolution/integration-scheme/). +This component belongs to the category of [integration schemes](../../../../simulation-principles/system-resolution/integration-scheme/). This scheme builds the system following an implicit scheme: forces are considered based on the state information at the next time step $x(t+dt)$, unknown at the current time step. -This scheme is an implicit time integrator for dynamic system using the Newmark scheme. To compute the new position or new velocity, the NewmarkIntegrationScheme is based on the following equations: +It is recommended to read the theoretical part of the [integration schemes](../../../../simulation-principles/system-resolution/integration-scheme/) page to understand the following. -$$ -x_{t+h}=x_t+h v_t+\frac{h^2}{2}((1-2\beta)a_t+2\beta a_{t+h}) -$$ - -$$ -v_{t+h}=v_t+h((1-\gamma)a_t+\gamma a_{t+h}) -$$ +The Newmark-$\beta$ integration scheme is a parametrized integration scheme. It presents a set of two parameters $\zeta = (\beta, \gamma)$, balancing the 'implicitness' of the solver: +1. With $\zeta =(0,0.5)$, the scheme is an explicit central difference scheme +2. With $\zeta =(0.25,0.5)$ it behaves like an average constant acceleration scheme +3. With $\zeta =(1/6,0.5)$ it behaves like a linear accelerations scheme +4. If $2 \beta \geq \gamma \geq 1/2$, then the Newmark-$\beta$ method is stable regardless of the size of the time-step +It inherits from AccelerationBAsedIntegrationScheme as only the acceleration integration is implicit. Its equations are the following : -Applied to a mechanical system where $\small Ma_t+(r_MM+r_KK)v_t+Kx_t=f_{ext}$, we need to solve the following system: - - -$$ -\tiny Ma_{t+h}+(r_MM+r_KK)v_{t+h}+Kx_{t+h}=f_{ext} -$$ - -$$ -\tiny Ma_{t+h}+(r_MM+r_KK)(v_t+h((1-\gamma)a_t+\gamma a_{t+h}))+K(x_t+hv_t+\frac{h^2}{2}((1-2\beta)a_t+2\beta a_{t+h}))=f_{ext} $$ +\begin{aligned} +g_{\boldsymbol{x}}^{(t,h)} &: \boldsymbol{v}, \boldsymbol{a} \mapsto \boldsymbol{x}_t + h \boldsymbol{v}_t + h^2\left[\left(\frac{1}{2} - \beta\right)\boldsymbol{a}_t + \beta \boldsymbol{a}\right] \\ +g_{\boldsymbol{v}}^{(t,h)} &: \boldsymbol{a} \mapsto \boldsymbol{v}_t + h \left[(1-\gamma)\boldsymbol{a}_t + \gamma \boldsymbol{a}\right] +\end{aligned} +$$ -$$ -\tiny (M+h\gamma(r_MM+r_KK)+h^2\beta K)a_{t+h}=f_{ext}-(r_MM+r_KK)(v_t+h(1-\gamma)a_t)-K(x_t+hv_t+\frac{h^2(1-2\beta)}{2}a_t) -$$ -$$ -\tiny ((1+h\gamma r_M)M+(h^2\beta +h\gamma r_K)K)a_{t+h}=f_{ext}-(r_MM+r_KK)v_t-Kx_t-(r_MM+r_KK)(h(1-\gamma)a_t)-K(hv_t+\frac{h^2(1-2\beta)}{2}a_t) -$$ +#### API specialization +As explained in the [IntegrationScheme](../../../../simulation-principles/system-resolution/integration-scheme/) documentation, the specialization consist in implementing 5 methods that requires the knowledge of four terms/expressions. For Newmark-$\beta$ they are the following : $$ -\tiny ((1+h\gamma r_M)M+(h^2\beta+h\gamma r_K)K)a_{t+h}=a_t-(r_MM+r_KK)(h(1-\gamma)a_t)-K(hv_t+\frac{h^2(1-2\beta)}{2}a_t) +\frac{\mathrm{d} g_{\boldsymbol{x}}^{(t,h)}}{\mathrm{d} \boldsymbol{v}} = 0 +\qquad \qquad \qquad +\frac{\mathrm{d} g_{\boldsymbol{x}}^{(t,h)}}{\mathrm{d} \boldsymbol{a}} = h^2\beta +\qquad \qquad \qquad +\frac{\mathrm{d} g_{\boldsymbol{v}}^{(t,h)}}{\mathrm{d} \boldsymbol{a}} = h \gamma $$ - - -Sequence diagram ----------------- - - Usage diff --git a/30_Components/40_IntegrationScheme/20_Backward/BDFIntegrationScheme.md b/30_Components/40_IntegrationScheme/20_Backward/BDFIntegrationScheme.md index 2a95fbddf..7379caf28 100644 --- a/30_Components/40_IntegrationScheme/20_Backward/BDFIntegrationScheme.md +++ b/30_Components/40_IntegrationScheme/20_Backward/BDFIntegrationScheme.md @@ -4,67 +4,66 @@ BDFIntegrationScheme This component belongs to the category of [integration schemes](../../../../simulation-principles/system-resolution/integration-scheme/). It is an implicit method for the numerical integration of the ODE resulting from Newton's second law of motion. -The method relies on [Backward Differentiation Formula](https://en.wikipedia.org/wiki/Backward_differentiation_formula) (BDF). +It is recommended to read the theoretical part of the [integration schemes](../../../../simulation-principles/system-resolution/integration-scheme/) page to understand the following. + +The method relies on [Backward Differentiation Formula](https://en.wikipedia.org/wiki/Backward_differentiation_formula) (BDF). This is a special case of [linear multistep method](https://en.wikipedia.org/wiki/Linear_multistep_method). To integrate the ODE in time, it uses information from the previous time steps to compute the next step. It establishes a linear combination of the unknown states, the previous states and the values of the ODE function when applied on those states. -This equation leads to a nonlinear function to solve. -That is why this component requires a [NewtonRaphsonSolver](NewtonRaphsonSolver.md), which the purpose is to solve nonlinear functions. -The coefficients of the linear combination come from the approximation of the function by a Lagrange interpolation polynomial. +It the specific case of BDF, the coefficients of the linear combination come from the approximation of the function by a Lagrange interpolation polynomial. The order of the BDF is the number of previous time steps required to approximate the interpolation polynomial. The first-order BDF requires a single time step in the past to compute the next. It corresponds to the [backward Euler method](EulerImplicitSolver.md). The coefficients are unique for a given order, but can be influenced by a change of time step size. The SOFA component supports any order, and any change of time step size. -Details -------- +In the following we are going to present the generic linear multistep method equations from which BFD derives. It has to be noted that we made the choice to only apply the linear multi step method to the position integration, not on the velocity. Using it to both decreases the stability margin of the integration scheme, leading to big instabilities in the scenes. -The ODE resulting from Newton's second law of motion is: +The LinearMultistepIntegrationScheme inherits from VelocityBaseIntegrationScheme because its integration scheme in velocity is invertible. Its equations are the following : $$ -\begin{bmatrix} -\frac{d q}{d t}\\ -M \frac{d \dot{q}}{d t} -\end{bmatrix} -= -\begin{bmatrix} -\dot{q}\\ -F(q, \dot{q}) -\end{bmatrix} +\begin{aligned} +g_{\boldsymbol{x}}^{(t,h)} &: \boldsymbol{v}, \boldsymbol{a} \mapsto - \sum_{j=1}^{n} \frac{\alpha_j}{\alpha_{n+1}} \boldsymbol{x}_{t-n+j} + h \frac{\beta_{n+1}}{\alpha_{n+1}} \boldsymbol{v} + h \sum_{j=1}^{n} \frac{\beta_j}{\alpha_{n+1}} \boldsymbol{v}_{t-n+j} \\ +g_{\boldsymbol{v}}^{(t,h)} &: \boldsymbol{a} \mapsto \boldsymbol{v}_{t} + h \boldsymbol{a} +\end{aligned} $$ -where $q$ and $\dot{q}$ are respectively the position and the velocity, $M$ is the mass matrix, and $F$ is the sum of forces. - -We define $y(t)=\begin{bmatrix} q \\ \dot{q} \end{bmatrix}$, and $f(t,y)=\begin{bmatrix} \dot{q} \\ M^{-1} F(q,\dot{q}) \end{bmatrix}$, such that the ODE is $y'=f(t,y)$. -We are interested in computing $y(t_{n+s})$ and we know the values of $y(t_{n+j})$ for $j \lt s$. -The Lagrange interpolation polynomial is the linear combination $L(t) = \sum_{j=0}^s y(t_n+j) l_j(t)$, where $l_j$ is the basis polynomials defined as $l_j(t)=\prod_{0 \leq m \leq s, m \neq j} \frac{t-t_{n+m}}{t_{n+j}-t{n+m}}$. -Then, we approximate $y'$ by $L'$, leading to the equation $\sum_{j=0}^s y_{n+j} l'_j(t_{n+s}) = f(t_{n+s},y_{n+s})$. -We can now define a nonlinear function $r(q, \dot{q}) = \left[\sum_{j=0}^s y_{n+j} l'_j(t_{n+s})\right] - f(t_{n+s},y_{n+s})$. -The roots of this function corresponds to $y_{tn+s}$, i.e. the next values of the state. +with $n$ the order of the integration scheme -To find the root of this function, we use a Newton-Raphson algorithm. -It means the derivative of the function is necessary. -At each iteration $i$ of the algorithm, we solve the following linear system: +#### API specialization +As explained in the [IntegrationScheme](../../../../simulation-principles/system-resolution/integration-scheme/) documentation, the specilization consist in implementing 4 methods that requires the knowledge of four terms/expressions. for Euler implicit they are the following : $$ -\begin{bmatrix} -l'_s(t_{n+s}) I & -dt I \\ --dt \frac{\partial F}{\partial q} & l'_s(t_{n+s}) M - dt \frac{\partial F}{\partial \dot{q}} -\end{bmatrix} -\begin{bmatrix} -q^{i+1} - q^i \\ -\dot{q}^{i+1} - \dot{q}^{i} -\end{bmatrix} -= -r(q^i, \dot{q}^i) +\begin{aligned} +&\tilde{g}_{\boldsymbol{x}}^{(t,h)} \equiv g_{\boldsymbol{x}}^{(t,h)} \\ +&g_{\boldsymbol{v}}^{(t,h)-1} : \boldsymbol{v} \mapsto \frac{\boldsymbol{v} - \boldsymbol{v}_t}{h} +\end{aligned} +\qquad \qquad \qquad +\begin{aligned} +&\frac{\mathrm{d} \tilde{g}_{\boldsymbol{x}}^{(t,h)}}{\mathrm{d} \boldsymbol{v}} = h \frac{\beta_{n+1}}{\alpha_{n+1}} \\ +&\frac{\mathrm{d} g_{\boldsymbol{v}}^{(t,h)-1}}{\mathrm{d} \boldsymbol{v}} = \frac{1}{h} +\end{aligned} $$ -This system is solved by a block Gaussian elimination leading to the velocity-based linear system: -$$ -\left(l'_s(t_{n+s}) M - dt \frac{\partial F}{\partial \dot{q}} - \frac{dt^2}{l'_s(t_{n+s})} \frac{\partial F}{\partial q} \right) -(\dot{q}^{i+1} - \dot{q}^i) -= -r_{\dot{q}}(q^i, \dot{q}^i) - \frac{dt}{l'_s(t_{n+s})} \frac{\partial F}{\partial q} r_{q}(q^i, \dot{q}^i) -$$ +#### BDF implementation + +The generic implementation of LinearMultistepIntegrationScheme requires to overide only one virtual funciton being : + +```cpp +// Method that will compute the $\alpha$ and $\beta factors$ +virtual void computeFactors() = 0; +``` + +The the implementation of BDFIntegrationScheme only computes those terms using the previously cited Lagrange polynomial interpolation. + + +----- + +The BDFIntegrationScheme **requires**: + +- a [LinearSolver](../../../../simulation-principles/system-resolution/linear-solver/) to solve the linear system +- and a MechanicalObject to store the state vectors. + From f2bd981da3fa691d10c0c31be9fb0f10cf4302ce Mon Sep 17 00:00:00 2001 From: Paul Baksic Date: Fri, 31 Jul 2026 11:58:51 +0200 Subject: [PATCH 10/10] Remove TODO link as they can be guessed --- .../40_System_Resolution/10_Integration_Scheme.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/20_Simulation_Principles/40_System_Resolution/10_Integration_Scheme.md b/20_Simulation_Principles/40_System_Resolution/10_Integration_Scheme.md index 65a7d8afc..862076ebb 100644 --- a/20_Simulation_Principles/40_System_Resolution/10_Integration_Scheme.md +++ b/20_Simulation_Principles/40_System_Resolution/10_Integration_Scheme.md @@ -408,7 +408,7 @@ virtual void computeCurrentPositionIntegrationError(...) = 0; virtual void computeCurrentVelocityIntegrationError(...) = 0; ``` -The tree first method returning only scalar values, they are the most traightforward method to implement. The two last have to deal with advanced concept of SOFA such as mechanical operation on `VecId`. For an example on how to implement this, see the Newmak implementation [here](//TODO, link to cpp file in the master branch once the PR is merged). +The tree first method returning only scalar values, they are the most traightforward method to implement. The two last have to deal with advanced concept of SOFA such as mechanical operation on `VecId`. For an example on how to implement this, see the Newmak implementation [here](https://github.com/sofa-framework/sofa/blob/master/Sofa/Component/IntegrationScheme/Backward/src/sofa/component/integrationscheme/backward/NewmarkIntegrationScheme.cpp). #### VelocityBasedIntegrationScheme @@ -438,7 +438,7 @@ virtual void computeCurrentPositionIntegrationError(...) = 0; //This method compute the acceleration given the current velocity, or $g_v^{(t,h)-1}$ virtual void computeAccelerationFromVelocity(...) = 0; ``` -Again, the two first method returning only scalar values, they are the most traightforward method to implement. The two last have to deal with advanced concept of SOFA such as mechanical operation on `VecId`. For an example on how to implement this, see the Euler implicit implementation [here](//TODO, link to cpp file in the master branch once the PR is merged). +Again, the two first method returning only scalar values, they are the most traightforward method to implement. The two last have to deal with advanced concept of SOFA such as mechanical operation on `VecId`. For an example on how to implement this, see the Euler implicit implementation [here](https://github.com/sofa-framework/sofa/blob/master/Sofa/Component/IntegrationScheme/Backward/src/sofa/component/integrationscheme/backward/EulerImplicitIntegrationScheme.cpp). > **Note :**\ > The velocity-base integration schemes offer a possibility to reduce the integration to a _first order_ integration, meaning the velocity is considered as null at the begining of each time step. This feature can help for quasi-static simulation or simulations where the objects dynamic is by nature subject to numerical noise such as very lightwheight objects. \