JSON api

Json files¶

Complete example

{ "mesh": " ", "bc_tag": " ", "boundary_id_threshold": -1, "normalize_mesh": true,

"n_refs": 0, "refinenemt_location": 0.5,

"scalar_formulation": "Laplacian""tensor_formulation": "LinearElasticity""count_flipped_els": false,

"iso_parametric": false, "discr_order": 1,

"pressure_discr_order": 1,

"output": "",

"problem": "Franke", "problem_params": {},           "n_boundary_samples": 6, "quadrature_order": 4,

"export": {                     "full_mat": "", "iso_mesh": "", "nodes": "", "solution": "", "spectrum": false, "stiffness_mat": "", "vis_boundary_only": false, "vis_mesh": "", "wire_mesh": "" },

"use_spline": false, "fit_nodes": false,

"integral_constraints": 2, "n_harmonic_samples": 10,

"B": 3, "use_p_ref": false, "discr_order_max": 4, "h1_formula": false,

"solver_type": "Pardiso", "precond_type": "Eigen::DiagonalPreconditioner&q "solver_params": {},            "nl_solver": "newton", "line_search": "armijo", "nl_solver_rhs_steps": 1, "save_solve_sequence": false,

"params": {                     "k": 1.0,

"elasticity_tensor": {},    "E": 1.5, "nu": 0.3,

"lambda": 0.329670329, "mu": 0.384615384, },

"tend": 1, "time_steps": 10,

"vismesh_rel_area": 1e-05       }

"Mesh path" "Path to the boundary tag file, each face/edge is associated with an unique number (you can use bc_setter for setting them in 3d)" "Distance from bounding box for a face/edge to belong to boundary. Negative falls into defaul: in 2d is 1e-7, in 3d 1e-2" "Normalize mesh such that it fits in the [0,1] bounding box" "Number of uniform refinement" "Refiniement location of polyhedra" class="p">, class="p">, "Count (or not) flipped elements" "Force isoparametric elements" "Dicretization order, supports P1, P2, P3, P4, Q1, Q2" "Pressure dicrezation order, for mixed formulation" "Output json path" "Problem to solve" class="nt">"Problem specific parameters" "number of boundary samples (Dirichelt) or quadrature points (Neumann)" "quadrature order" class="nt">"Export options" "Stiffnes matrix without boundary conditions" "Isolines mesh" "FE nodes" "Solution vector" "Spectrum of the stiffness matrix" "Stiffmess matrix after setting boundary conditions" "Exports only the boundary of volumetric meshes" "Path for the vtu mesh" "Wireframe of the mesh" "Use spline for quad/hex elements" "Fit nodes for spline basis" "Number of constraints for polygonal basis 0, 1, or 2" "Number of face/line samples for harmonic bases for polyhedra" "User provided parameter for pref tolerance" "Enable prefinement for badly shaped simplices" "Maximum allowed dicrezation oder, used in p pref" "Use pref formula for h1 bound" "Library for linear solver" uot;, class="nt">"solver specific parameters" "Non linear solver" "Line search for newton solver" "Number of incremental load steps" "Save all incremental load steps" class="nt">"Material parameter" "Constant in helmolz" class="nt">"Elasticity tensor, used in hooke and saint ventant" "Young modulus" "Poisson's ratio" "Lame parameter, E, nu have priority" "End time for time dependent simulations" "Number of time steps for time dependent simulations" class="s2">"Relative resolution of the output mesh"

Optionals¶

scalar_formulation: Helmholtz, Laplacian, Bilaplacian (mixed)
tensor_formulation: HookeLinearElasticity, LinearElasticity, NeoHookean, SaintVenant, IncompressibleLinearElasticity (mixed), Stokes (mixed)
problem: CompressionElasticExact, Cubic, DrivenCavity, Elastic, ElasticExact, ElasticZeroBC, Flow, Franke, GenericScalar, GenericTensor, Gravity, Kernel, Linear, LinearElasticExact, MinSurf, PointBasedTensor, Quadratic, QuadraticElasticExact, Sine, TestProblem, TimeDependentFlow, TimeDependentScalar, TorsionElastic, Zero_BC
solver_type: Eigen::BiCGSTAB, Eigen::ConjugateGradient, Eigen::GMRES, Eigen::MINRES, Eigen::SimplicialLDLT, Eigen::SparseLU, Hypre,Pardiso
nl_solver: lbfgs, newton
line_search: armijo, armijo_alt, bisection, more_thuente

Formulations¶

List of possible formulations. The constants can be set in params. All formulations supports boundary conditions.

Scalar¶

Laplacian¶

Constants: none
Description: solve for
$-\Delta u = f$

Bilaplacian (mixed)¶

Constants: none
Description: solve for
$-\Delta^2 u = f$

Helmholtz¶

Constants: $k$
Description: solve for
$-\Delta u - k^2 u = f$

Tensor¶

LinearElasticity¶

Constants: young/nu, E/nu, lambda/nu
Description: solve for
$-\text{div}(\sigma[u]) = f \qquad \sigma[u] = 2 \mu \epsilon[u]+ \lambda \text{tr}(\epsilon[u]) I \qquad \epsilon[u] = \frac 1 2 \left(\nabla u^T + \nabla u\right)$

HookeLinearElasticity¶

Constants: elasticity_tensor, young/nu, E/nu, lambda/nu
Description: solve for
$-\text{div}(\sigma[u]) = f \qquad \sigma[u] = C : \epsilon[u] \qquad \epsilon[u] = \frac 1 2 \left(\nabla u^T + \nabla u\right)$
where $C$ is the elasticity tensor

IncompressibleLinearElasticity (mixed)¶

Constants: young/nu, E/nu, lambda/nu
Description: solve for
$\begin{align} -\text{div}(2\mu\epsilon[u] + p I) &= f\\ \text{div}(u) - \lambda^{-1}p &= 0 \end{align}$

SaintVenant¶

Constants: elasticity_tensor, young/nu, E/nu, lambda/nu
Description: solve for
$-\text{div}(\sigma[u]) = f \qquad \sigma[u] = C: \epsilon[u] \qquad \epsilon[u] = \frac 1 2 \left(\nabla u^T \nabla u + \nabla u^T + \nabla u\right)$
where $C$ is the elasticity tensor

NeoHookean¶

Constants: young/nu, E/nu, lambda/nu
Description: solve for
$-\text{div}(\sigma[u]) = f \quad \sigma[u] = \mu (F[u] - F[u]^{-T}) + \lambda \ln(\det F[u]) F[u]^{-T} \quad F[u] = \nabla u + I$

Stokes (mixed)¶

Constants: viscosity $\mu$
Description: solve for
$\begin{align} -\mu\Delta u + \nabla p &= f\\ -\text{div}(u) &= 0 \end{align}$

Problems¶

Each problem has a specific set of optional problem_params described here.

CompressionElasticExact¶

Has exact solution: true
Time dependent: false
Form: tensor
Description: solve for
$\begin{align} f_{2D}(x,y) &= -\begin{bmatrix}(y^3 + x^2 + xy)/20\\ (3x^4 + xy^2 + x)/20\end{bmatrix}\\ f_{3D}(x,y,z) &= -\begin{bmatrix}(xy + x^2 + y^3 + 6z)/14\\ (zx - z^3 + xy^2 + 3x^4)/14\\ (xyz + y^2z^2 - 2x)/14\end{bmatrix} \end{align}$

Cubic¶

Has exact solution: true
Time dependent: false
Form: scalar
Description: solve for
$f(x,y,z) = (2y-0.9)^4 + 0.1$

DrivenCavity¶

Has exact solution: false
Time dependent: false
Form: tensor
Description: solve for zero right-hand side, and 0.25 for boundary id 1

Elastic¶

Has exact solution: false
Time dependent: false
Form: tensor
Description: solve for zero right-hand side, -0.25 for boundary id ⅕, 0.25 for id 3/6

ElasticExact¶

Has exact solution: true
Time dependent: false
Form: tensor
Description: solve for
$\begin{align} f_{2D}(x,y) &= \begin{bmatrix}(y^3 + x^2 + xy)/50\\ (3x^4 + xy^2 + x)/50\end{bmatrix}\\ f_{3D}(x,y,z) &= \begin{bmatrix}(xy + x^2 + y^3 + 6z)/80\\ (xz - z^3 + xy^2 + 3x^4)/80\\ (xyz + y^2 z^2 - 2x)/80\end{bmatrix} \end{align}$

ElasticZeroBC¶

Has exact solution: false
Time dependent: false
Form: tensor
Description: solve for [0, 0.5, 0] right-hand side and zero boundary condition

Flow¶

Has exact solution: false
Time dependent: false
Form: tensor
Description: solve for zero right-hand side, [0.25, 0, 0] for boundary id ⅓, [0, 0, 0] for 7

Franke¶

Has exact solution: true
Time dependent: false
Form: scalar
Description: solves for the 2D and 3D Franke function

GenericScalar¶

Has exact solution: false
Time dependent: false
Form: scalar
Description: solves for generic tensor problem with zero rhs
Options:

"rhs": 3                            "Rhs of the problem"
"dirichlet_boundary": [             "List of Dirichelt boundaries"
{
    "id": 1,                        "Boundary id"
    "value": 0                      "Boundary value"
},
{
    "id": 2,                        "Boundary id"
    "value": "sin(x)+y"             "Formulas are supported"
}],
"neumann_boundary": [               "List of Neumann boundaries"
{
    "id": 3,                        "Boundary id"
    "value": 1,                     "Boundary value"
},
{
    "id": 4,                        "Boundary id"
    "value": "x^2"                  "Formulas are supported"
}]

GenericTensor¶

Has exact solution: false
Time dependent: false
Form: tesor
Description: solves for generic tensor problem with zero body forces
Options:

"rhs": [1, 2, 3]                    "Rhs of the problem"
"dirichlet_boundary": [             "List of Dirichelt boundaries"
{
    "id": 1,                        "Boundary id"
    "value": [0, 0, 0],             "Boundary vector value"
    "dimension": [                  "Which dimension are Dirichelt"
            true,
            true,
            false                   "In this case z is free"
        ]
},
{
    "id": 2,                        "Boundary id"
    "value": ["sin(x)+y", "z^2", 0] "Formulas are supported"
}],
"neumann_boundary": [               "List of Neumann boundaries"
{
    "id": 3,                        "Boundary id"
    "value": [0, 0, 0]              "Boundary vector value"
},
{
    "id": 4,                        "Boundary id"
    "value": ["sin(z)+y", "z^2", 0] "Formulas are supported"
}]

Gravity¶

Has exact solution: false
Time dependent: true
Form: tensor
Description: solves for 0.1 body force in y direction and zeor for boundray 4

Kernel¶

Has exact solution: true
Time dependent: false
Form: scalar/tensor
Description: solves the omogenous PDE with n_kernels kernels placed on the bounding box at kernel_distance
Options: n_kernels sets the number of kernels, kernel_distance sets the distance from the bounding box

Linear¶

Has exact solution: true
Time dependent: false
Form: scalar
Description: solve for
f(x,y,z) = x

LinearElasticExact¶

Has exact solution: true
Time dependent: false
Form: tensor
Description: solve for
$\begin{align} f_{2D}(x,y) &= \begin{bmatrix}-(y + x)/50\\ -(3x + y)/50\end{bmatrix}\\ f_{3D}(x,y,z) &= \begin{bmatrix}-(y + x + z)/50\\ -(3x + y - z)/50\\ -(x + y - 2z)/50\end{bmatrix}\\ \end{align}$

MinSurf¶

Has exact solution: false
Time dependent: false
Form: scalar
Description: solve for -10 for rhs, and zero Dirichelt boundary condition

PointBasedTensor¶

Has exact solution: false
Time dependent: false
Form: tesor
Description: solves for point-based boudary conditions
Options:

"scaling": 1,               "Scaling factor"
"rhs": 0,                   "Right-hand side"
"translation": [0, 0, 0]    "Translation"
"boundary_ids": [           "List of Dirichelt boundaries"
{
    "id": 1,                "Boundary id"
    "value": [0, 0, 0]      "Boundary vector value"
},
{
    "id": 2,
    "value": {              "Rbf interpolated value"
        "function": "",     "Function file"
        "points": "",       "Points file"
        "rbf": "gaussian",  "Rbf kernel"
        "epsilon": 1.5,     "Rbf epsilon"
        "coordinate": 2,    "Coordinate to ignore"

        "dimension": [      "Which dimension are Dirichelt"
            true,
            true,
            false           "In this case z is free"
        ]
    }
},
,
{
    "id": 2,
    "value": {              "Rbf interpolated value"
        "function": "",     "Function file"
        "points": "",       "Points file"
        "triangles": "",    "Triangles file"
        "coordinate": 2,    "Coordinate to ignore"
    }
}]

Quadratic¶

Has exact solution: true
Time dependent: false
Form: scalar
Description: solve for
$f(x,y,z) = x^2$

QuadraticElasticExact¶

Has exact solution: true
Time dependent: false
Form: tensor
Description: solve for
$\begin{align} f_{2D}(x,y) &= \begin{bmatrix} -(y^2 + x^2 + xy)/50\\ -(3x^2 + y)/50\end{bmatrix}\\ f_{3D}(x,y,z) &= \begin{bmatrix}-(y^2 + x^2 + xy + yz)/50\\ -(3x^2 + y + z^2)/50\\ -(xz + y^2 - 2z)/50\end{bmatrix} \end{align}$

Sine¶

Has exact solution: true
Time dependent: false
Form: scalar
Description: solve for
$\begin{align} f(x,y) &= \sin(10x)\sin(10y)\\ f(x,y,z) &= \sin(10x)\sin(10y)\sin(10z) \end{align}$

TestProblem¶

Has exact solution: true
Time dependent: false
Form: scalar
Description: solve for extreme problem to test errors for high order discretizations

TimeDependentFlow¶

Has exact solution: false
Time dependent: true
Form: tensor
Description: solve for zero right-hand side, [0.25, 0, 0] for boundary id ⅓, [0, 0, 0] for 7, and zero inital velocity

TimeDependentScalar¶

Has exact solution: false
Time dependent: true
Form: scalar
Description: solve for one right-hand side, zero boundary condition, and zero time boundary

TorsionElastic¶

Has exact solution: false
Time dependent: false
Form: tensor
Description: solve for zero body forces, fixed_boundary fixed (zero displacement), turning_boundary rotating around axis_coordiante for n_turns
Options: fixed_boundary id of the fixed boundary, turning_boundary id of the moving boundary, axis_coordiante coordinate of the rotating axis, n_turns number of turns

Zero_BC¶

Has exact solution: true
Time dependent: false
Form: tensor
Description: solve for
$\begin{align} f_{2D}(x,y) &= (1 - x) x^2 y (1-y)^2\\ f_{3D}(x,y,z) &= (1 - x) x^2 y (1-y)^2 z (1 - z) \end{align}$