pyxfem.solidmechanics.exact_solutions module

This file contains exact solutions to some classic pde problems

Use full to test convergence Created on Wed Feb 28 09:49:32 2024 @author: nchevaug

class pyxfem.solidmechanics.exact_solutions.LaplacianCircularInclusionQuad(k1, k2, ri=0.4, re=np.float64(1.4142135623730951))

Bases: LaplacianGeneral

Circular inclusion, with quadratic solution

u equal zero at re, source term ‘s = 1.’ is constant, 2 conductivity : k1 inside, k2 outside

class pyxfem.solidmechanics.exact_solutions.LaplacianCircularInclusionQuintic(k1, k2, ri=0.4, re=np.float64(1.4142135623730951))

Bases: LaplacianGeneral

Circular inclusion with order 5 displacement source (r-re)**3

class pyxfem.solidmechanics.exact_solutions.LaplacianGeneral(lsfuncs=None, classifyer=None, filt=0, conductivity=None, source=None, BNDDefinition=None, DirichletBCs=None, NeumannBCs=None, exactSolution=None, exactGradient=None)

Bases: object

Base class for the definition of Laplacian type problem and its exact solution

get_conductivity_val()

return a dictionary of conductivity values

get_dirichlet_bc_val()

return a dictionary of boundary name to dirichlet value

get_exact_grad()

return the gradient exact solution of the problem

get_exact_sol()

return the exact solution of the problem

get_ls()

get The level-set function used to define the problem (when multiple conductiviies are present )

get_neumann_bc_val()

return a dictionary of boundary name to neumann value

get_source_term()

return the souce term associated to the problem

class pyxfem.solidmechanics.exact_solutions.LaplacianSquareHole(a=0.4)

Bases: LaplacianGeneral

Manufactured solution : square with a square hole

+–+ | | +–+

class pyxfem.solidmechanics.exact_solutions.LinearElasticity2dCircularHoleInfiniteSpace(mat: LinearIsotropeElasticMaterial, r=0.5, p=1.0)

Bases: LinearElasticityExacSol

linear Elasticity exact solution : hole in a plate

ddf(z)

fist derivative of the complex potential f evaluation

ddg(z)

second derivative of the complex potential g evaluation

df(z)

fist derivative of the complex potential f evaluation

dg(z)

fist derivative of the complex potential g evaluation

displacement(xy)

return the displacement value at each position xy

f(z)

complex potential f evaluation

strain(xy, dim=2)

return the displacement field

if dim = 2 return array shape is (..,2,2) if dim = 3 return array shape is (..,3,3)

stress(xy, dim=2)

return the stress field

if dim = 2 return array shape is (..,2,2) if dim = 3 return array shape is (..,3,3)

class pyxfem.solidmechanics.exact_solutions.LinearElasticityExacSol

Bases: ABC

Base class for exact solution of linear elasticity problems

abstractmethod displacement(xy)

Return the displacement at each point in the xy input array input array of float such as […, :] contain the x and y coordinate of the point. return an array of the sameshape containing the displacement at the input point

abstractmethod strain(xy)

Return the displacement at each point in the xy input array input array of float such as […, :] contain the x and y coordinate of the point. return an array of shape xy.shape[:-1] + (2,2) containing the 2d strain.

abstractmethod stress(xy)

Return the displacement at each point in the xy input array input array of float such as […, :] contain the x and y coordinate of the point. return an array of shape xy.shape[:-1] + (2,2) containing the 2d stress.

class pyxfem.solidmechanics.exact_solutions.WilliamsExpansion(mat, A=None, B=None)

Bases: object

Exact field from williams expansion (crack tip is at (0,0), x axis aligned with the crack)

^ y | |

______o—> x crack

displacement(xy)

Compute the displacement at xy

strain(xy)

Compute the stress at xy

stress(xy)

Compute the stress at xy

pyxfem.solidmechanics.exact_solutions.elasticity_hole_in_plate(popUpGmsh=False)

test exactSolutions.linearElasticity2dCircularHoleInfiniteSpace and plot it

pyxfem.solidmechanics.exact_solutions.laplacian_inclusion(exactsolbuilder, popUpGmsh=False)

plot exact solution for laplacien with 2 conductivities on a mesh

class pyxfem.solidmechanics.exact_solutions.testAnaly(methodName='runTest')

Bases: TestCase

testing Analytical solutions

testElasticityHoleInPlate(popUpGmsh=False)

• Compute max exact stress on mesh nodes and compare to ref value

testLaplacianCircularInclusionQuad(popUpGmsh=False)

• Compute max Temperature on mesh nodes and compare to ref value

testLaplacianCircularInclusionQuintic(popUpGmsh=False)

• Compute max Temperature on mesh nodes and compare to ref value