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Problems

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

Generic

GenericScalar

  • Has exact solution: false
  • Time-dependent: false
  • Form: scalar
  • Description: solves for generic scalar problem with specified rhs
  • Options: 
    "rhs": 3                            // Rhs of the problem
"dirichlet_boundary": [             // List of Dirichlet 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: user-selected
  • Form: tensor
  • Description: solves for generic tensor problem with specified 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
}],
"is_time_dependent": false,
"initial_solution": [0, 0, 0],
"initial_velocity": [0, 0, 0],
"initial_acceleration": [0, 0, 0]

Specific

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

Gravity

  • Has exact solution: false
  • Time-dependent: true
  • Form: tensor
  • Description: solves for 0.1 body force in y direction and zero for boundary 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: tensor
  • 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 Dirichlet
            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
\[\begin{align} f_{2D}(x,y) &= (1 - x) x^2 y (1-y)^2\\ f_{3D}(x,y,z) &= (1 - x) x^2 y (1-y)^2 z (1 - z) \end{align}\]
Last update: 2023-10-03