Euler Explicit ode solver solves the following \(2^{nd}\) order ode \(M(x(t), v(t)).a(t) = f(t, x(t), v(t))\). More...

Inheritance diagram for SurgSim::Math::OdeSolverEulerExplicit:

Public Member Functions
	OdeSolverEulerExplicit (OdeEquation *equation)
	Constructor. More...

void	solve (double dt, const OdeState &currentState, OdeState *newState, bool computeCompliance=true) override
	Solves the equation. More...

Public Member Functions inherited from SurgSim::Math::OdeSolver
	OdeSolver (OdeEquation *equation)
	Constructor. More...

virtual	~OdeSolver ()
	Virtual destructor.

const std::string	getName () const
	Gets the solver's name. More...

void	setLinearSolver (std::shared_ptr< LinearSparseSolveAndInverse > linearSolver)
	Sets the specialized linear solver to use with this Ode solver. More...

std::shared_ptr< LinearSparseSolveAndInverse >	getLinearSolver () const
	Gets the specialized linear solver used with this Ode solver. More...

void	computeMatrices (double dt, const OdeState &state, bool computeCompliance=true)
	Computes the system and compliance matrices for a given state. More...

const SparseMatrix &	getSystemMatrix () const
	Queries the current system matrix. More...

const Matrix &	getComplianceMatrix () const

Protected Member Functions
void	assembleLinearSystem (double dt, const OdeState &state, const OdeState &newState, bool computeRHS=true) override
	Assemble the linear system (A.x=b) to be solved for the state and new states (useful for certain ode solver). More...

Protected Member Functions inherited from SurgSim::Math::OdeSolver
void	computeComplianceMatrixFromSystemMatrix (const OdeState &state)
	Helper method computing the compliance matrix from the system matrix and setting the boundary conditions. More...

Additional Inherited Members
Protected Attributes inherited from SurgSim::Math::OdeSolver
std::string	m_name
	Name for this solver. More...

OdeEquation &	m_equation
	The ode equation (API providing the necessary evaluation methods and the initial state)

std::shared_ptr< LinearSparseSolveAndInverse >	m_linearSolver
	The specialized linear solver to use when solving the ode equation.

SparseMatrix	m_systemMatrix
	Linear system matrix (can be M, K, combination of MDK depending on the solver), including boundary conditions. More...

Vector	m_solution
	Linear system solution and rhs vectors (including boundary conditions)

Vector	m_rhs

Matrix	m_complianceMatrix
	Compliance matrix which is the inverse of the system matrix, including boundary conditions.

Detailed Description

Euler Explicit ode solver solves the following \(2^{nd}\) order ode \(M(x(t), v(t)).a(t) = f(t, x(t), v(t))\).

This ode is solved as an ode of order 1 by defining the state vector \(y = \left(\begin{array}{c}x\\v\end{array}\right)\):

\[ y' = \left(\begin{array}{c} x' \\ v' \end{array}\right) = \left(\begin{array}{c} v \\ M(x, v)^{-1}.f(t,x, v) \end{array}\right) = f(t, y) \]

After integrating this equation, we get:

\[ y(t+dt) - y(t) = \int_t^{t+dt} f(t,y) dt \]

Note: Euler explicit uses a rectangular numerical integration on the left to evaluate this integral, leading to \( \int_t^{t+dt} f(t,y) dt \simeq dt.f(t, y(t))\), therefore:
\[ \begin{array}{ccc} y(t+dt) - y(t) = dt.f(t, y(t)) & \Leftrightarrow & \left\{ \begin{array}{ccccl} x(t+dt) &=& x(t) &+& dt.v(t) \\ v(t+dt) &=& v(t) &+& dt.a(t) \end{array} \right. \end{array} \]; Euler Explicit is also known as forward Euler as it uses a forward evaluation of the derivative \(y' = (y(t+dt) - y(t)) / dt\) which leads to the same result.

Constructor & Destructor Documentation

SurgSim::Math::OdeSolverEulerExplicit::OdeSolverEulerExplicit ( OdeEquation * equation )

explicit

Constructor.

Parameters

equation The ode equation to be solved

void SurgSim::Math::OdeSolverEulerExplicit::assembleLinearSystem	(	double	dt,
		const OdeState &	state,
		const OdeState &	newState,
		bool	computeRHS = `true`
	)

overrideprotectedvirtual

Assemble the linear system (A.x=b) to be solved for the state and new states (useful for certain ode solver).

Parameters

dt	The time step used in the system
state,newState	The state and newState to be used to evaluate the system
computeRHS	True to compute the RHS vector, False otherwise

Note: The method should fill up the LHS matrix in m_systemMatrix and the RHS vector in m_b (if requested); The method should take care of the boundary conditions properly on both the matrix and the vector.; The method should prepare the linear solver m_linearSolver to be used with the m_systemMatrix

void SurgSim::Math::OdeSolverEulerExplicit::solve	(	double	dt,
		const OdeState &	currentState,
		OdeState *	newState,
		bool	computeCompliance = `true`
	)

overridevirtual

Solves the equation.

Parameters

	dt	The time step
	currentState	State at time t
[out]	newState	State at time t+dt
	computeCompliance	True to explicitly compute the compliance matrix, False otherwise

The documentation for this class was generated from the following files: