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Documentation

Json files

Complete example

{
    "mesh": " ",                    "Mesh path"
    "bc_tag": " ",                  "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)"
    "boundary_id_threshold": -1,    "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": true,         "Normalize mesh such that it fits in the [0,1] bounding box"

    "n_refs": 0,                    "Number of uniform refinement"
    "refinenemt_location": 0.5,     "Refiniement location of polyhedra"

    "scalar_formulation": "Laplacian",
    "tensor_formulation": "LinearElasticity",

    "count_flipped_els": false,     "Count (or not) flipped elements"

    "iso_parametric": false,        "Force isoparametric elements"
    "discr_order": 1,               "Dicretization order, supports P1, P2, P3, P4, Q1, Q2"

    "pressure_discr_order": 1,      "Pressure dicrezation order, for mixed formulation"

    "output": "",                   "Output json path"

    "problem": "Franke",            "Problem to solve"
    "problem_params": {},           "Problem specific parameters"

    "n_boundary_samples": 6,        "number of boundary samples (Dirichelt) or quadrature points (Neumann)"
    "quadrature_order": 4,          "quadrature order"

    "export": {                     "Export options"
        "full_mat": "",             "Stiffnes matrix without boundary conditions"
        "iso_mesh": "",             "Isolines mesh"
        "nodes": "",                "FE nodes"
        "solution": "",             "Solution vector"
        "spectrum": false,          "Spectrum of the stiffness matrix"
        "stiffness_mat": "",        "Stiffmess matrix after setting boundary conditions"
        "vis_boundary_only": false, "Exports only the boundary of volumetric meshes"
        "vis_mesh": "",             "Path for the vtu mesh"
        "wire_mesh": ""             "Wireframe of the mesh"
    },

    "use_spline": false,            "Use spline for quad/hex elements"
    "fit_nodes": false,             "Fit nodes for spline basis"

    "integral_constraints": 2,      "Number of constraints for polygonal basis 0, 1, or 2"
    "n_harmonic_samples": 10,       "Number of face/line samples for harmonic bases for polyhedra"

    "B": 3,                         "User provided parameter for pref tolerance"
    "use_p_ref": false,             "Enable prefinement for badly shaped simplices"
    "discr_order_max": 4,           "Maximum allowed dicrezation oder, used in p pref"
    "h1_formula": false,            "Use pref formula for h1 bound"

    "solver_type": "Pardiso",       "Library for linear solver"
    "precond_type": "Eigen::DiagonalPreconditioner",
    "solver_params": {},            "solver specific parameters"

    "nl_solver": "newton",          "Non linear solver"
    "line_search": "armijo",        "Line search for newton solver"
    "nl_solver_rhs_steps": 1,       "Number of incremental load steps"
    "save_solve_sequence": false,   "Save all incremental load steps"

    "params": {                     "Material parameter"
        "k": 1.0,                   "Constant in helmolz"

        "elasticity_tensor": {},    "Elasticity tensor, used in hooke and saint ventant"

        "E": 1.5,                   "Young modulus"
        "nu": 0.3,                  "Poisson's ratio"

        "lambda": 0.329670329,      "Lame parameter, E, nu have priority"
        "mu": 0.384615384,
    },


    "tend": 1,                      "End time for time dependent simulations"
    "time_steps": 10,               "Number of time steps for time dependent simulations"

    "vismesh_rel_area": 1e-05       "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
$\begin{align} f_{2D}(x,y) &= (1 - x) x^2 y (1-y)^2\\ f_{3D}(x,y,z) &= (1 - x) x^2 y (1-y)^2 z (1 - z) \end{align}$