PDESolverOptions Properties

Algorithm options for solvers

A PDESolverOptions object contains options used by the solvers when solving a structural, thermal, or general PDE problem specified as a StructuralModel, ThermalModel, or PDEModel object, respectively. StructuralModel, ThermalModel, and PDEModel objects contain a PDESolverOptions object in their SolverOptions property.

Solvers for structural modal analysis problems and reduced-order modeling use the Lanczos algorithm.

Statistics and Convergence Report

ReportStatistics — Flag to display internal solver statistics and convergence report during the solution process

'off' (default) | 'on'

Flag to display the internal solver statistics and the convergence report during the solution process, returned as 'off' or 'on'.

Example: model.SolverOptions.ReportStatistics = 'on'

Data Types: char

ODE Solver

AbsoluteTolerance — Absolute tolerance for internal ODE solver

1.0000e-06 (default) | positive number

Absolute tolerance for the internal ODE solver, returned as a positive number. Absolute tolerance is a threshold below which the value of the solution component is unimportant. This property determines the accuracy when the solution approaches zero.

Example: model.SolverOptions.AbsoluteTolerance = 5.0000e-06

Data Types: double

RelativeTolerance — Relative tolerance for internal ODE solver

1.0000e-03 (default) | positive number

Relative tolerance for the internal ODE solver, returned as a positive number. This tolerance is a measure of the error relative to the size of each solution component. Roughly, it controls the number of correct digits in all solution components, except those smaller than thresholds imposed by AbsoluteTolerance. The default value corresponds to 0.1% accuracy.

Example: model.SolverOptions.RelativeTolerance = 5.0000e-03

Data Types: double

Nonlinear Solver

ResidualTolerance — Acceptable residual tolerance for internal nonlinear solver

1.0000e-04 (default) | positive number

Acceptable residual tolerance for the internal nonlinear solver, returned as a positive number. The nonlinear solver iterates until the residual size is less than the value of ResidualTolerance.

Example: model.SolverOptions.ResidualTolerance = 5.0000e-04

Data Types: double

MaxIterations — Maximal number of Gauss-Newton iterations for internal nonlinear solver

25 (default) | positive integer

Maximal number of Gauss-Newton iterations for the internal nonlinear solver, returned as a positive integer.

Example: model.SolverOptions.MaxIterations = 30

Data Types: double

MinStep — Minimum damping of search direction for internal nonlinear solver

1.5259e-05 (default) | positive number

Minimum damping of the search direction for the internal nonlinear solver, returned as a positive number. For details, see Nonlinear Solver Algorithm.

Example: model.SolverOptions.MinStep = 1.5259e-7

Data Types: double

ResidualNorm — Type of norm for computing residual for internal nonlinear solver

Inf (default) | -Inf | positive number | 'energy'

Type of norm for computing the residual for the internal nonlinear solver, returned as Inf, -Inf, a positive number, or 'energy'.

The infinity norms of a vector are

$${\Vert \rho \Vert }_{\infty }={\max }_i\left(\left|\rho \left(i\right)\right|\right)$$

$${\Vert \rho \Vert }_{-\infty }={\min }_i\left(\left|\rho \left(i\right)\right|\right)$$

The Lp-norm of a vector ρ that has N elements is

$${\Vert \rho \Vert }_p=\frac{{\left[{\displaystyle \sum _{k=1}^N{\left|{\rho }_k\right|}^p}\right]}^{\frac{1}{p}}}{N^{\frac{1}{p}}}$$

The energy norm of a vector ρ is

$$\Vert \rho \Vert ={\rho }^{T}K\rho$$

Here, K is the combined stiffness matrix defined in Nonlinear Solver Algorithm.

Example: model.SolverOptions.ResidualNorm = 'energy'

Data Types: double | char

Lanczos Solver

MaxShift — Maximum number of Lanczos shifts

100 (default) | positive integer

Maximum number of Lanczos shifts, specified as a positive integer. Increase this value when computing a large number of eigenpairs.

Example: model.SolverOptions.MaxShift = 500

Data Types: double

BlockSize — Block size for block Lanczos recurrence

ranges from 7 to 25 (default) | positive integer

Block size for block Lanczos recurrence, specified as a positive integer. The default number ranges from 7 to 25, depending on the size of the stiffness matrix K.

Example: model.SolverOptions.BlockSize = 20

Data Types: double

Algorithms

Nonlinear Solver Algorithm

The residual equation of a nonlinear PDE is as follows:

$$r\left(u\right)=-\nabla \cdot \left(c\left(u\right)\nabla \left(u\right)\right)+a\left(u\right)u-f\left(u\right)=0$$

To obtain a discretized residual equation, apply the finite element method (FEM) to a partial differential equation as described in Finite Element Method Basics:

$$\rho \left(U\right)=K\left(U\right)U-F\left(U\right)=0$$

The nonlinear solver uses a Gauss-Newton iteration scheme applied to the finite element matrices. Use a Taylor series expansion to obtain the linearized system for the residual:

$$\rho \left(U^{n+1}\right)\cong \rho \left(U^n\right)+\frac{\partial \rho \left(U^n\right)}{\partial U}\left(U^{n+1}-U^n\right)+\dots =0$$

Neglecting the higher-order terms, write the linearized system of equations as

$$\frac{\partial \rho \left(U^n\right)}{\partial U}\left(U^{n+1}-U^n\right)=-\rho \left(U^n\right)$$

The descent direction for the residual is

$$p_n=-{\left(\frac{\partial \rho \left(U^n\right)}{\partial U}\right)}^{-1}\rho \left(U^n\right)$$

The Gauss-Newton iteration minimizes the residual, that is, the solution of $${\min }_U\Vert \rho \left(U\right)\Vert$$, using the equation

$$U^{n+1}=U^n+\alpha p_n$$

Here, ɑ ≤ 1 is a positive number, that must be set as large as possible so that the step has a reasonable descent. For a sufficiently small ɑ,

$$\Vert \rho \left(U^n+\alpha p_n\right)\Vert <\Vert \rho \left(U^n\right)\Vert$$

For the Gauss-Newton algorithm to converge, $$U^0$$ must be close enough to the solution. The first guess is often outside the region of convergence. The Armijo-Goldstein line search (a damping strategy for choosing ɑ) helps to improve convergence from bad initial guesses. This method chooses the largest damping coefficient ɑ out of the sequence 1, 1/2, 1/4, . . . such that the following inequality holds:

$$\Vert \rho \left(U^n\right)\Vert -\Vert \rho \left(U^n+\alpha p_n\right)\Vert \ge \frac{\alpha }{2}\Vert \rho \left(U^n\right)\Vert$$

Using the Armijo-Goldstein line search guarantees a reduction of the residual norm by at least $$1-\alpha /2$$. Each step of the line-search algorithm must evaluate the residual $$\Vert \rho \left(U^n+\alpha p_n\right)\Vert$$.

With this strategy, when Uⁿ approaches the solution, $$\alpha$$→1, thus, the convergence rate increases.