valid3D module¶

Contents

Mass Matrix
Stiffness matrix
Elastic Stiffness Matrix

Mass Matrix ¶

Validation program for the assembly of the Mass matrix for $P_1$ -Lagrange finite element method in 3d (see Mass Matrix). Meshes of the unit cube are used.

Test 1: Computation of the Mass Matrix using the base, OptV1 and OptV2 versions : it gives errors and computation times

Test 2: Computation of the integral

$\int_{\Omega_h} u(\q) v(\q) d\q \approx \pmb{V}^t\mathbb{M} \pmb{U}$

where $\pmb{U}_i=u(q^i)$ and $\pmb{V}_i=v(q^i)$ . Functions $u$ and $v$ are those defined in ...

Test 3: Ones retrieves the order 2 of $P_1$ -Lagrange integration

$|\int_{\Omega_h} u\,v -\Pi_h(u)\,\Pi_h(v)d\Omega| \leq C h^2$

Note

source code

pyOptFEM.valid3D.validMass3DP1.validMass3DP1(**kwargs)[source]: Validation of Mass Matrix for P1-Lagrange finite elements in 3D

>>> from pyOptFEM.valid3D import validMass3DP1
>>> validMass3DP1()
*********************************************
*     3D Mass Assembling P1 validations     *
*********************************************
-----------------------------------------
  Test 1: Matrices errors and CPU times
-----------------------------------------
    Matrix size           : (216,216)
    Error P1base vs OptV1 : 1.734723e-18
    Error P1base vs OptV2 : 1.734723e-18
    CPU times base (ref)  : 0.9355 (s)
    CPU times OptV1       : 0.0481 (s) - Speed Up X19.438
    CPU times OptV2       : 0.0016 (s) - Speed Up X592.083
----------------------------
  Test 1 (results): OK
----------------------------
-----------------------------------------------------
  Test 2: Validations by integration on [0,1]x[0,1]
-----------------------------------------------------
    function 0 :
      u(x,y,z)=x + y + z,
      v(x,y,z)=x - y - z,
           -> Stiff error=3.330669e-16
    function 1 :
      u(x,y,z)=3*x + 2*y - z - 1,
      v(x,y,z)=2*x - 2*y + 2*z + 1,
           -> Stiff error=8.881784e-16
    function 2 :
      u(x,y,z)=3*x**2 - x*y + 2*y**2 + y*z - z**2 - 3,
      v(x,y,z)=2*x**2 + x*y - 3*y**2 - x*z - y,
           -> Stiff error=1.173244e-02
----------------------------
  Test 2 (results): OK
----------------------------
--------------------------------
  Test 3: Validations by order
--------------------------------
    function 2 :
      u(x,y,z)=3*x**2 - x*y + 2*y**2 + y*z - z**2 - 3,
      v(x,y,z)=2*x**2 + x*y - 3*y**2 - x*z - y
        Matrix size                     : (216,216)
        MassAssemblingP1OptV2 CPU times : 0.002(s)
        -> Error                         : 1.173244e-02
        Matrix size                     : (1331,1331)
        MassAssemblingP1OptV2 CPU times : 0.009(s)
        -> Error                         : 2.800865e-03
        ...
        Matrix size                     : (132651,132651)
        MassAssemblingP1OptV2 CPU times : 1.418(s)
        -> Error                         : 1.110034e-04

At last, this program plots the figure :

Figure 59: Graphical results of validMass3DP1-Test 3

Stiffness matrix ¶

Validation function for the assembly of the Stiffness matrix for $P_1$ -Lagrange finite element method in 3d (see Stiffness Matrix). Meshes of the unit cube are used.

Test 1: Computation of the Stiffness Matrix using the base, OptV1 and OptV2 versions : it gives errors and computation times

Test 2: Computation of the integral

$\int_{\Omega_h} \nabla u(q).\nabla v(q)dq \approx \pmb{V}^t\mathbb{M} \pmb{U}$

where $\pmb{U}_i=u(q^i)$ and $\pmb{V}_i=v(q^i)$ . Functions $u$ and $v$ are those defined in ...

Test 3: Ones retrieves the order 2 of $P_1$ -Lagrange integration

$|\int_{\Omega_h} \nabla u . \nabla v -\nabla\Pi_h(u) . \nabla\Pi_h(v)d\Omega| \leq C h^2$

Note

source code

pyOptFEM.valid3D.validStiff3DP1.validStiff3DP1(**kwargs)[source]: Validation of Stiffness Matrix for P1-Lagrange finite elements in 3D

>>> from pyOptFEM.valid3D import validStiff3DP1
>>> validStiff3DP1()
**********************************************
*     3D Stiff Assembling P1 validations     *
**********************************************
-----------------------------------------
  Test 1: Matrices errors and CPU times
-----------------------------------------
    Matrix size           : (216,216)
    Error P1base vs OptV1 : 2.220446e-15
    Error P1base vs OptV2 : 2.220446e-15
    CPU times base (ref)  : 0.9563 (s)
    CPU times OptV1       : 0.0566 (s) - Speed Up X16.901
    CPU times OptV2       : 0.0032 (s) - Speed Up X299.656
----------------------------
  Test 1 (results): OK
----------------------------
-----------------------------------------------------
  Test 2: Validations by integration on [0,1]x[0,1]
-----------------------------------------------------
    function 0 :
      u(x,y,z)=x + y + z,
      v(x,y,z)=x - y - z,
           -> Stiff error=6.217249e-15
    function 1 :
      u(x,y,z)=3*x + 2*y - z - 1,
      v(x,y,z)=2*x - 2*y + 2*z + 1,
           -> Stiff error=7.771561e-15
    function 2 :
      u(x,y,z)=3*x**2 - x*y + 2*y**2 + y*z - z**2 - 3,
      v(x,y,z)=2*x**2 + x*y - 3*y**2 - x*z - y,
           -> Stiff error=2.077333e-02
----------------------------
  Test 2 (results): OK
----------------------------
--------------------------------
  Test 3: Validations by order
--------------------------------
    function 2 :
      u(x,y,z)=3*x**2 - x*y + 2*y**2 + y*z - z**2 - 3,
      v(x,y,z)=2*x**2 + x*y - 3*y**2 - x*z - y
        Matrix size                        : (216,216)
        StiffAssembling3DP1OptV2 CPU times : 0.003(s)
        Error                              : 2.077333e-02
        Matrix size                        : (1331,1331)
        StiffAssembling3DP1OptV2 CPU times : 0.019(s)
        Error                              : 3.330000e-03
        ...
        Matrix size                        : (132651,132651)
        StiffAssembling3DP1OptV2 CPU times : 2.784(s)
        Error                              : 1.828928e-04

At last, this program plots the figure :

Figure 60: Graphical results of validStiff3DP1-Test 3

Elastic Stiffness Matrix ¶

Validation function for the assembly of the Elastic Stiffness matrix for $P_1$ -Lagrange finite element method in 3d (see Elastic Stiffness Matrix). Meshes of the unit cube are used.

Test 1: Computation of the Elastic Stiffness Matrix using the base, OptV1 and OptV2 versions : it gives errors and computation times

Test 2: Computation of the integral

$\int_{\Omega_h} \underline{\pmb{\epsilon}}^t(u) \underline{\pmb{\sigma}}(v)dq \approx \pmb{V}^t\mathbb{M} \pmb{U}$

where $\pmb{U}_i=u(q^i)$ and $\pmb{V}_i=v(q^i)$ . Functions $u$ and $v$ are those defined in ...

Test 3: Ones retrieves the order 2 of $P_1$ -Lagrange integration

$|\int_{\Omega_h} \underline{\pmb{\epsilon}}^t(u) \underline{\pmb{\sigma}}(v) - \underline{\pmb{\epsilon}}^t(\Pi_h(u)) \underline{\pmb{\sigma}}(\Pi_h(v)) d\Omega| \leq C h^2$

Note

pyOptFEM.valid3D.validStiff3DP1.validStiffElas3DP1([Num=0, la=1.5, mu=0.5])¶

Parameters:	Num – (optional) Numbering choice. la – (optional) the first Lame coefficient in Hooke’s law, denoted by $\lambda$ . mu – (optional) the second Lame coefficient in Hooke’s law, denoted by $\mu$ .

>>> from pyOptFEM.valid3D import validStiffElas3DP1
>>> validStiffElas3DP1()
**************************************************
*     3D StiffElas Assembling P1 validations     *
**************************************************
-----------------------------------------
  Test 1: Matrices errors and CPU times
-----------------------------------------
    Matrix size           : (648,648)
    Error P1base vs OptV1 : 1.554312e-15
    Error P1base vs OptV2 : 1.776357e-15
    CPU times base (ref)  : 2.8850 (s)
    CPU times OptV1       : 0.2925 (s) - Speed Up X9.862
    CPU times OptV2       : 0.0152 (s) - Speed Up X189.919
----------------------------
  Test 1 (results): OK
----------------------------
-----------------------------------------------------
  Test 2: Validations by integration on [0,1]x[0,1]
-----------------------------------------------------
    functions 0 :
        u(x,y,z)=['x - 2*y', 'x + y - z', '3*x + 2*z'],
        v(x,y,z)=['x + 2*y + 4*z', '2*x - y + 4*z', '3*x - 2*y'],
           -> StiffElas error=5.329071e-14
    functions 1 :
        u(x,y,z)=['5*x - 2*y+z', 'x + y - 3*z', '3*x + -2*y+ 2*z'],
        v(x,y,z)=['2*x - 2*y + 4*z +1', '5*x - y + 4*z', '4*x - 2*y+4'],
           -> StiffElas error=2.131628e-14
    functions 2 :
        u(x,y,z)=['x**2 - 2*x*y + x*z', 'y**2 - y*z + z**2 + x', 'x**2 - x*z - y*z - z**2'],
        v(x,y,z)=['x**2 + 2*y**2 - x*z', '2*x**2 - x*y + y*z', 'x*y - y*z + z**2'],
           -> StiffElas error=5.712000e-02
    functions 3 :
        u(x,y,z)=['x**2 - 2*x*y + x*z', 'x**3 + y**2 - y*z + z**2', '-x**2*z - x*y*z + x**2 - z**2'],
        v(x,y,z)=['-x*z**2 + x**2 + 2*y**2', '2*x**2 - x*y + y*z', 'x*y - y*z + z**2'],
           -> StiffElas error=1.221381e-01
----------------------------
  Test 2 (results): OK
----------------------------
--------------------------------
  Test 3: Validations by order
--------------------------------
    functions 3 :
        u(x,y,z)=[x**2 - 2*x*y + x*z,x**3 + y**2 - y*z + z**2,-x**2*z - x*y*z + x**2 - z**2],
        v(x,y)=[-x*z**2 + x**2 + 2*y**2,2*x**2 - x*y + y*z,x*y - y*z + z**2]
      Matrix size                            : (3993,3993)
      StiffElasAssembling3DP1OptV2 CPU times : 0.121(s)
      -> Error                               : 3.036230e-02
      Matrix size                            : (12288,12288)
      StiffElasAssembling3DP1OptV2 CPU times : 0.422(s)
      -> Error                               : 1.515389e-02
      Matrix size                            : (27783,27783)
      StiffElasAssembling3DP1OptV2 CPU times : 0.999(s)
      -> Error                               : 8.314471e-03
      Matrix size                            : (52728,52728)
      StiffElasAssembling3DP1OptV2 CPU times : 2.019(s)
      -> Error                               : 5.618296e-03
      Matrix size                            : (89373,89373)
      StiffElasAssembling3DP1OptV2 CPU times : 3.528(s)
      -> Error                               : 3.666314e-03
      Matrix size                            : (139968,139968)
      StiffElasAssembling3DP1OptV2 CPU times : 5.668(s)
      -> Error                               : 2.932322e-03

At last, this program plots the figure :

Figure 61: Graphical results of validStiffElas3DP1-Test 3

valid3D module¶

Mass Matrix ¶

Stiffness matrix ¶

Elastic Stiffness Matrix ¶

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valid3D module¶

Mass Matrix¶

Stiffness matrix¶

Elastic Stiffness Matrix¶

Table Of Contents

Previous topic

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Mass Matrix ¶

Stiffness matrix ¶

Elastic Stiffness Matrix ¶