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

$$\begin{array}{rl}{f}_{2D}(x,y)& =-\left[\begin{array}{c}(y^3+x^2+xy)/20\\ (3x^4+x{y}^2+x)/20\end{array}\right]\\ f_{3D}(x,y,z)& =-\left[\begin{array}{c}(xy+x^2+y^3+6z)/14\\ (zx-z^3+x{y}^2+3x^4)/14\\ (xyz+y^2{z}^2-2x)/14\end{array}\right]\end{array}$$

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

$$\begin{array}{rl}{f}_{2D}(x,y)& =\left[\begin{array}{c}(y^3+x^2+xy)/50\\ (3x^4+x{y}^2+x)/50\end{array}\right]\\ f_{3D}(x,y,z)& =\left[\begin{array}{c}(xy+x^2+y^3+6z)/80\\ (xz-z^3+x{y}^2+3x^4)/80\\ (xyz+y^2{z}^2-2x)/80\end{array}\right]\end{array}$$

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

$$\begin{array}{rl}{f}_{2D}(x,y)& =\left[\begin{array}{c}-(y+x)/50\\ -(3x+y)/50\end{array}\right]\\ f_{3D}(x,y,z)& =\left[\begin{array}{c}-(y+x+z)/50\\ -(3x+y-z)/50\\ -(x+y-2z)/50\end{array}\right]\end{array}$$

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

$$\begin{array}{rl}{f}_{2D}(x,y)& =\left[\begin{array}{c}-(y^2+x^2+xy)/50\\ -(3x^2+y)/50\end{array}\right]\\ f_{3D}(x,y,z)& =\left[\begin{array}{c}-(y^2+x^2+xy+yz)/50\\ -(3x^2+y+z^2)/50\\ -(xz+y^2-2z)/50\end{array}\right]\end{array}$$

Sine

• Has exact solution: true
• Time-dependent: false
• Form: scalar
• Description: solve for

$$\begin{array}{rl}f(x,y)& =\sin \left(10x\right)\sin \left(10y\right)\\ f(x,y,z)& =\sin \left(10x\right)\sin \left(10y\right)\sin \left(10z\right)\end{array}$$

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

$$\begin{array}{rl}{f}_{2D}(x,y)& =(1-x)x^{2}y(1-y{)}^2\\ f_{3D}(x,y,z)& =(1-x)x^{2}y(1-y{)}^{2}z(1-z)\end{array}$$

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Last update: 2023-10-03