valid2D module¶

Contents

Mass Matrix
Stiffness matrix
Elastic Stiffness Matrix

Mass Matrix ¶

Validation program for the assembly of the Mass matrix $\MAT{M}$ for $P_1$ -Lagrange finite element method in 2d (see Mass Matrix).

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

$\int_{\Omega_h} u(x,y) v(x,y) dxdy \approx \pmb{V}^t\mathbb{M} \pmb{U}$

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.valid2D.validMass2DP1.validMass2DP1(**kwargs)[source]¶: Validation of Mass Matrix for P1-Lagrange finite elements in 2D

>>> from pyOptFEM.valid2D import validMass2DP1
>>> validMass2DP1()
******************************************
*     Mass Assembling P1 validations     *
******************************************
-----------------------------------------
Test 1: Matrices errors and CPU times
-----------------------------------------
    Matrix size           : (121,121)
    Error P1base vs OptV1 : 8.673617e-19
    Error P1base vs OptV2 : 8.673617e-19
    CPU times base (ref)  : 0.2638 (s)
    CPU times OptV1       : 0.0259 (s) - Speed Up X10.179
    CPU times OptV2       : 0.0016 (s) - Speed Up X160.286
----------------------------
  Test 1 (results): OK
----------------------------
-----------------------------------------------------
  Test 2: Validations by integration on [0,1]x[0,1]
-----------------------------------------------------
    function 0 : u(x,y)=x+2*y, v(x,y)=3*x+y+1,
            -> Mass error=0.000000e+00
    function 1 : u(x,y)=x**2+2*y*x+y, v(x,y)=3*x*y+y**2+1,
            -> Mass error=5.736389e-03
    function 2 : u(x,y)=x**3+2*y**2*x+y**2+x, v(x,y)=2*x*y+y**3+x*y,
            -> Mass error=1.141472e-02
----------------------------
  Test 2 (results): OK
----------------------------
--------------------------------
  Test 3: Validations by order
--------------------------------
  functions 2: u(x,y)=x**3+2*y**2*x+y**2+x, v(x,y)=2*x*y+y**3+x*y
      Matrix size                     : (121,121)
      MassAssemblingP1OptV2 CPU times : 0.001(s)
      Error                           : 1.141472e-02
      Matrix size                     : (441,441)
      MassAssemblingP1OptV2 CPU times : 0.002(s)
      Error                           : 2.825605e-03
      ...
      Matrix size                     : (10201,10201)
      MassAssemblingP1OptV2 CPU times : 0.019(s)
      Error                           : 1.125457e-04

At last, this program plots the figure :

Figure 56: Graphical results of validMass2DP1-Test 3

Stiffness matrix ¶

Validation function for the assembly of the Stiffness matrix, $\MAT{S}$ , for $P_1$ -Lagrange finite element method in 2d (see Stiffness Matrix)

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

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

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.valid2D.validStiff2DP1.validStiff2DP1(**kwargs)[source]: validation of stiffness matrix assembling functions

>>> from pyOptFEM.valid2D import validStiff2DP1
>>> validStiff2DP1()
**********************************************
*     2D Stiff Assembling P1 validations     *
**********************************************
-----------------------------------------
  Test 1: Matrices errors and CPU times
-----------------------------------------
    Matrix size           : (121,121)
    Error P1base vs OptV1 : 8.881784e-16
    Error P1base vs OptV2 : 4.440892e-16
    CPU times base (ref)  : 0.1519 (s)
    CPU times OptV1       : 0.0106 (s) - Speed Up X14.270
    CPU times OptV2       : 0.0009 (s) - Speed Up X174.238
----------------------------
  Test 1 (results): OK
----------------------------
-----------------------------------------------------
  Test 2: Validations by integration on [0,1]x[0,1]
-----------------------------------------------------
    function 0 : u(x,y)=x+2*y, v(x,y)=3*x+y+1,
          -> Stiff error=2.486900e-14
    function 1 : u(x,y)=x**2+2*y*x+y, v(x,y)=3*x*y+y**2+1,
          -> Stiff error=2.000000e-02
    function 2 : u(x,y)=x**3+2*y**2*x+y**2+x, v(x,y)=2*x*y+y**3+x*y,
          -> Stiff error=1.500000e-02
----------------------------
  Test 2 (results): OK
----------------------------
--------------------------------
  Test 3: Validations by order
--------------------------------
Test 2: u(x,y)=x**3+2*y**2*x+y**2+x, v(x,y)=2*x*y+y**3+x*y
      Matrix size                      : (121,121)
      StiffAssemblingP1OptV2 CPU times : 0.001(s)
      Error                            : 1.500000e-02
      ...
      Matrix size                      : (10201,10201)
      StiffAssemblingP1OptV2 CPU times : 0.021(s)
      Error                            : 1.500000e-04

At last, this program plots this figure :

Figure 57: Graphical results of validStiff2DP1-Test 3

Elastic Stiffness Matrix ¶

Validation function for the assembly of the Elastic Stiffness matrix, $\MAT{K}$ , for $P_1$ -Lagrange finite element method in 2d (see Elastic Stiffness Matrix)

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

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

where $\vecb{u}=(u_1,u_2)$ and $\vecb{v}=(v_1,v_2)$ are given 2d-vector functions and $\pmb{U}$ and $\pmb{V}$ are the vectors in $\R^{2\nq}$ defined by

if Num=0 or Num=2 (i.e. in global alternate basis $\mathcal{B}_a$ , see Assembly matrices (base, OptV1 and OptV2 versions) )

$\forall i\in\{1,\hdots,\nq\},\ U_{2i-1}=u_1(\q^i),\ U_{2i}=u_2(\q^i),\ \mbox{ and } \ V_{2i-1}=v_1(\q^i),\ \ V_{2i}=v_2(\q^i)$

if Num=1 or Num=3 (i.e. in global block basis $\mathcal{B}_b$ , see Assembly matrices (base, OptV1 and OptV2 versions) )

$\forall i\in\{1,\hdots,\nq\},\ U_{i}=u_1(\q^i),\ U_{i+\nq}=u_2(\q^i),\ \mbox{ and } \ V_{i}=v_1(\q^i),\ \ V_{i+\nq}=v_2(\q^i)$

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.valid2D.validStiff2DP1.validStiffElas2DP1([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 $\mu$ .

pyOptFEM.valid2D.validStiffElas2DP1.validStiffElas2DP1(**kwargs)[source]: validation of elasticity stiffness matrix assembly functions

>>> from pyOptFEM.valid2D import validStiffElas2DP1
>>> validStiffElas2DP1()
**************************************************
*     2D StiffElas Assembling P1 validations     *
**************************************************
  Numbering Choice : global alternate numbering with local alternate numbering
-----------------------------------------
  Test 1: Matrices errors and CPU times
-----------------------------------------
    Matrix size           : (1922,1922)
    Error P1base vs OptV1 : 1.776357e-15
    Error P1base vs OptV2 : 2.664535e-15
    CPU times base (ref)  : 1.9251 (s)
    CPU times OptV1       : 0.2817 (s) - Speed Up X6.835
    CPU times OptV2       : 0.0098 (s) - Speed Up X195.725
----------------------------
  Test 1 (results): OK
----------------------------
-----------------------------------------------------
  Test 2: Validations by integration on [0,1]x[0,1]
-----------------------------------------------------
    function 0 :
      u(x,y)=[x - 2*y,x + y],
      v(x,y)=[x + 2*y,2*x - y],
           -> StiffElas error=8.704149e-14
    function 1 :
      u(x,y)=[x**2+2*y*x+y,-2*y**2+x**2+x-y],
      v(x,y)=[3*x*y+y**2+1,3*x**2-x*y+1],
           -> StiffElas error=3.916049e-03
    function 2 :
      u(x,y)=[x**3+2*y**2*x+y**2+x,y**3-2*x**2*y],
      v(x,y)=[2*x*y+y**3+x*y,3*x**3-2*x*y+x-1],
           -> StiffElas error=6.574856e-03
----------------------------
  Test 2 (results): OK
----------------------------
--------------------------------
  Test 3: Validations by order
--------------------------------
    functions 2:
      u(x,y)=['x**3+2*y**2*x+y**2+x', 'y**3-2*x**2*y'],
      v(x,y)=['2*x*y+y**3+x*y', '3*x**3-2*x*y+x-1'],
      lambda=1.500000, mu=0.500000
        Matrix size                          : (242,242)
        StiffElasAssemblingP1OptV2 CPU times : 0.002(s)
        Error                                : 5.767667e-02
        Matrix size                          : (882,882)
        StiffElasAssemblingP1OptV2 CPU times : 0.005(s)
        Error                                : 1.447542e-02
        Matrix size                          : (1922,1922)
        ...
        Matrix size                          : (20402,20402)
        StiffElasAssemblingP1OptV2 CPU times : 0.110(s)
        Error                                : 5.797263e-04

At last, this program plots this figure :

Figure 58: Graphical results of validStiffElas2DP1-Test 3

valid2D module¶

Mass Matrix ¶

Stiffness matrix ¶

Elastic Stiffness Matrix ¶

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valid2D module¶

Mass Matrix¶

Stiffness matrix¶

Elastic Stiffness Matrix¶

Table Of Contents

Previous topic

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Mass Matrix ¶

Stiffness matrix ¶

Elastic Stiffness Matrix ¶