Generic one element tests

The purposes of the generic one-element tests are:

to verify the correctness of the automatically generated code when ported on various machines and for various finite element environments
to verify objectivity with respect to the superimposed rigid body motion on a deformed state of the element
to verify objectivity with respect to the translation and rotation of the reference coordinate system
to apprise behaviour of the element in constrained conditions as material uncompressibility and extremely distorted or elongated element shapes
to determine convergence characteristics of the element for the standard Newton-Raphson iterative scheme

Most of the data that characterize finite element such as node coordinates and boundary conditions is independent on a specific element formulation. That data is depicted below for all types of elements. Some data depends on a type of the element. That data can be found in a preamble of the element source code and on the element home page. In general we have the following element dependent data:

boundary condition multipliers (or load factors) for which analysis was performed are presented on element home page
order of numerical integration (see integration code) is specified on the element home page
characteristic values of the element specific constants (material parameters, tolerances, etc.) are presented in a preamble of the element source code

The following standardized one element tests are performed:

generic one element test
test of the element eigenvalues and objectivity

Description of the results displayed on the element page

Generic one element test of code correctness

This is one element test with the arbitrary geometry, element constants and boundary conditions. It gives precise numerical values which can be used for checking the correctness of the element code when is ported on various machines an translated to various computer languages. The results of the code verification test can be found on an element home page. Results are given for all load factors l₁^...l_N. On element home page the following data is displayed:

both main diagonals of the global tangent matrix and residuum vector at the converge state for the prescribed load factor

K_i,i	K_1,1	K_2,2	...	K_NDOF,_NDOF
K_i,NDOF-i+1	K_NDOF,1	K_NDOF-1,2	...	K_1,NDOF
R_i	R₁	R₂	...	R_NDOF

solution vector (including prescribed boundary conditions)

a₁

a₂

a_NDOF

for all Newton-Raphson iterations the sum of the norm of the residual and the norm of the solution increment. This is the test of the element convergence behaviour.

||₁Y||+||D₁a||

||₂Y||+||D₂a||

...

sensitivity of the solution vector with respect to all chosen parameters p_i

da₁/dp_i

da₂/dp_i

da_NDOF/dp_Np

Sensitivity parameters p₁ ... p_Npa specified in a preamble of the element source code. In general parameters p₁ ... p_Np-1 represent material parameters and p_Npis the shape parameter that coresponds to the change of the x coordinate in a second node. The sensitivity analysis is based on algorithems for exact analytical differentiation of all equations involved. Analytical sensitivity analysis is independently verified by comparition with the finite difference method. Results of finite difference method are summarized in a table:

sensitivity parameter \ load multiplier	l₁	l₂	_...	l_N
p₁	da₃/dp₁	da₃/dp₁	_...	da₃/dp₁
p₂	da₃/dp₂	da₃/dp₂	_...	da₃/dp₂
...	...	_...	_...	_...
p_Np	da₃/p_Np	da₃/p_Np	...	da₃/p_Np

Results are given for all sensitivity parameters and all load factors l₁^...l_N, but only for the first component of the solution vector in a second node (da₃/dp_i). Finite difference approximation is expressed as

da₃/dp_i=(a₃(1.00000001 p_i)-a₃(p_i))/(0.00000001 p_i).

Tests of the element eigenvalues and objectivity

Two material states are taken into account:

generic material with material characteristics taken from the generic one element test
uncompressible material with Poisson's ratio set to 0.4999 and other material characteristics taken from the generic one element test

Two types of objectivity are considered:

objectivity with respect to the superimposed rigid body motion on a deformed state of the element
objectivity with respect to the translation and rotation of the reference coordinate system. In this case rigid body like motion is superimposed on the initial nodal coordinates

Test is performed for two geometry's:

undistorted
moderately distorted

On element home page the following data is displayed:

Geometry: Material: Load factor: f_c: f_b

u_T

v_T

w_T

u_Tref

v_Tref

w_Tref

a₁

a₂

...

a_NDOF

where Geometry can be distorted or undistorted, Material can be generic or uncompressible, Load factor is the boundary conditions multiplier, f_cis the angle of rotation of the reference coordinate system, f_bis the angle of superimposed rigid body rotation, {u_T,v_T,w_T} is the actual displacement vector in node T, {u_Tref,v_Tref,w_Tref} is the displacement vector in node T transformed back to the reference geometry and free of rigid body rotations, and a_i eigenvalues of the element tangent matrix. For steady ste problems one load factor is considered and for transient problems load factors that characterise all characteristic states of element (e.g elastic, plastic).

The following combinations are considered for geometricaly linear formulations:

	Geometry	Material	f_c	f_b
1	undistorted	uncompressible	0	0
2	distorted	uncompressible	0	0
3	undistorted	generic	0	0
4	undistorted	generic	f_c	0
5	distorted	generic	0	0
6	distorted	generic	f_c	0

The element pass the test on objectivity with respect to the rotation of the reference frame if the combinations 3,4, for undistorted case and 5,6 for distorted case exibit the same {u_Tref,v_Tref,w_Tref} displacement vector.

The following combinations are considered for geometricaly non-linear formulations:

	Geometry	Material	f_c	f_b
1	undistorted	uncompressible	0	0
2	distorted	uncompressible	0	0
3	undistorted	generic	0	0
4	undistorted	generic	f_c	0
5	undistorted	generic	0	f_b
6	undistorted	generic	f_c	f_b
7	distorted	generic	0	0
8	distorted	generic	f_c	0
9	distorted	generic	0	f_b
10	distorted	generic	f_c	f_b

The element passes both objectivity tests if the combinations 3,4,5,6 for undistorted case and 7,8,9,10 for distorted case exibit the same { u_Tref, v_Tref, w_Tref} displacement vector.

The element is considered as volumetric locking free element if the number of eigenvalues larger than 10⁵ in the combinations 1,2 is less or equal 2.

T1 Generic one element two dimensional test for arbitrary three node element

Geometric data and boundary conditions

	Coordinates		Prescribed displacement		Prescribed force
Node	X	Y	u	v	F_x	F_y
1	-0.5	0.1	0.015	-0.12	F_x1	F_y1
2	1.2	-0.3	u₂	0.02	10	F_y2
3	1.1	1.3	0	0	F_x3	F_y3

Initial configuration sensitivity used for the shape sensitivity analysis

Node	dX/dp	dY/dp
1	0	0
2	1	0
3	0	0

Geometric data and boundary conditions for tests of the element eigenvalues and objectivity - undistorted element shape

	Coordinates		Prescribed displacement		Prescribed force
Node	X	Y	u	v	F_x	F_y
1	0	0	0.0015	-0.0012
2	1	0	0.005	0.003
3^(T)	0.5	0.8660254037844386			0.01	-0.05

Geometric data and boundary conditions for tests of the element eigenvalues and objectivity - moderate distorted element shape

	Coordinates		Prescribed displacement		Prescribed force
Node	X	Y	u	v	F_x	F_y
1	0	0	0.0015	-0.0012
2	1	0	0.005	0.003
3^(T)	1.5	0.5			0.01	-0.05

Description of the results displayed on an element page

T2 Generic one element tests for 6 node triangle

Geometric data and boundary conditions for generic test

	Coordinates		Prescribed displacement		Prescribed force
Node	X	Y	u	v	Fx	Fy
1	0.5	0.1	0.015	-0.12	-	-
2	2.2	-0.3	-	0.02	10	-
3	2.1	1.3	0	0	-	-
4	1.5	0.	-	-	-	-
5	2.3	0.7	-	-	-	-
6	1.0	0.8	-	-	-	-

Geometric data and boundary conditions for tests of the element eigenvalues and objectivity - undistorted element shape

	Coordinates		Prescribed displacement		Prescribed force
Node	X	Y	u	v	Fx	Fy
1	0.	0.	0.0015	-0.0012	-	-
2	1.	0.	0.005	0.003	-	-
3	0.5	0.866025	-	-	0.01	-0.05
4	0.5	0	-	-	-	-
5	0.75	0.433013	-	-	-	-
6	0.25	0.433013	-	-	-	-

Geometric data and boundary conditions for tests of the element eigenvalues and objectivity - moderate distorted element shape

	Coordinates		Prescribed displacement		Prescribed force
Node	X	Y	u	v	Fx	Fy
1	0.	0.	0.0015	-0.0012	-	-
2	1.	0.	0.005	0.003	-	-
3	1.5	0.5	-	-	0.01	-0.05
4	0.5	0	-	-	-	-
5	1.2	0.25	-	-	-	-
6	0.75	0.2	-	-	-	-

Q1 Generic one element two dimensional tests for arbitrary four node element

Geometric data and boundary conditions for generic test

	Coordinates		Prescribed displacement		Prescribed force
Node	X	Y	u	v	F_x	F_y
1	-0.5	0.1	0.015	-0.12	F_x1	F_y1
2	1.2	-0.3	u₂	0.02	105	F_y2
3	1.1	1.3	u₃	0.015	0	F_y3
4	0.1	1.4	0.021	v₄	F_x4	0

Initial configuration sensitivity used for the shape sensitivity analysis

Node	dX/dp	dY/dp
1	0	0
2	1	0
3	0	0
4	0	0

Geometric data and boundary conditions for tests of the element eigenvalues and objectivity - undistorted element shape

	Coordinates		Prescribed displacement		Prescribed force
Node	X	Y	u	v	F_x	F_y
1	0	0	0.0001	-0.0002
2	1	0	0.0007	0.0001
3	1	1			0.0015	0.002
4^(T)	0	1

Geometric data and boundary conditions for tests of the element eigenvalues and objectivity - moderate distorted element shape

	Coordinates		Prescribed displacement		Prescribed force
Node	X	Y	u	v	F_x	F_y
1	-0.5	0.1	0.0001	-0.0002
2	1.2	-0.3	0.0007	0.0001
3	1.1	1.3			0.0015	0.002
4^(T)	0.1	1.4

Description of the results displayed on an element page

Q2 Generic one element test for arbitrary 2D 8 node element

Geometric data and boundary conditions for generic test

	Coordinates		Prescribed displacement		Prescribed force
Node	X	Y	u	v	Fx	Fy
1	0.5	0.1	0.015	-0.12	-	-
2	1.2	0.3	-	0.02	105	-
3	1.1	1.3	-	0.015	-	-
4	0.1	1.4	0.021	-	-	-
5	0.8	0.2	-	-	-	-
6	1.1	0.7	-	-	-	-
7	0.8	1.3	-	-	-	-
8	0.3	0.7	-	-	-	-

Geometric data and boundary conditions for tests of the element eigenvalues and objectivity - undistorted element shape

	Coordinates		Prescribed displacement		Prescribed force
Node	X	Y	u	v	Fx	Fy
1	0.	0.	0.0001	-0.0002	-	-
2	1.	0.	0.0007	0.0001	-	-
3	1.	1.	-	-	0.0015	0.002
4	0.	1.	-	-	-	-
5	0.5	0	-	-	-	-
6	1	0.5	-	-	-	-
7	0.5	1	-	-	-	-
8	0	0.5	-	-	-	-

Geometric data and boundary conditions for tests of the element eigenvalues and objectivity - moderate distorted element shape

	Coordinates		Prescribed displacement		Prescribed force
Node	X	Y	u	v	Fx	Fy
1	0.5	0.1	0.0001	-0.0002	-	-
2	1.2	0.3	0.0007	0.0001	-	-
3	1.1	1.3	-	-	0.0015	0.002
4	0.1	1.4	-	-	-	-
5	0.8	0.2	-	-	-	-
6	1.1	0.7	-	-	-	-
7	0.8	1.3	-	-	-	-
8	0.3	0.7	-	-	-	-

AX Q1 Generic one element axisymmetric tests for arbitrary four node element

Geometric data and boundary conditions

	Coordinates		Prescribed displacement		Prescribed force
Node	X	Y	u	v	F_x	F_y
1	0.5	0.1	0.015	-0.12	F_x1	F_y1
2	1.2	0.3	u₂	0.02	105	F_y2
3	1.1	1.3	u₃	0.015	0	F_y3
4	0.1	1.4	0.021	v₄	F_x4	0

Initial configuration sensitivity used for the shape sensitivity analysis

Node	dX/dp	dY/dp
1	0	0
2	1	0
3	0	0
4	0	0

Geometric data and boundary conditions for tests of the element eigenvalues and objectivity - undistorted element shape

	Coordinates		Prescribed displacement		Prescribed force
Node	X	Y	u	v	F_x	F_y
1	0	0	0.0001	-0.0002
2	1	0	0.0007	0.0001
3	1	1			0.0015	0.002
4^(T)	0	1

Geometric data and boundary conditions for tests of the element eigenvalues and objectivity - moderate distorted element shape

	Coordinates		Prescribed displacement		Prescribed force
Node	X	Y	u	v	F_x	F_y
1	0.5	0.1	0.0001	-0.0002
2	1.2	0.3	0.0007	0.0001
3	1.1	1.3			0.0015	0.002
4^(T)	0.1	1.4

Description of the results displayed on an element page

AX T1 Generic one element axisymmetric test for arbitrary three node element

Geometric data and boundary conditions

	Coordinates		Prescribed displacement		Prescribed force
Node	X	Y	u	v	F_x	F_y
1	0.5	0.1	0.015	-0.12	F_x1	F_y1
2	1.2	0.3	u₂	0.02	10	F_y2
3	1.1	1.3	0	0	F_x3	F_y3

Initial configuration sensitivity used for the shape sensitivity analysis

Node	dX/dp	dY/dp
1	0	0
2	1	0
3	0	0

Geometric data and boundary conditions for tests of the element eigenvalues and objectivity - undistorted element shape

	Coordinates		Prescribed displacement		Prescribed force
Node	X	Y	u	v	F_x	F_y
1	0	0	0.0015	-0.0012
2	1	0	0.005	0.003
3^(T)	0.5	0.8660254037844386			0.01	-0.05

Geometric data and boundary conditions for tests of the element eigenvalues and objectivity - moderate distorted element shape

	Coordinates		Prescribed displacement		Prescribed force
Node	X	Y	u	v	F_x	F_y
1	0	0	0.0015	-0.0012
2	1	0	0.005	0.003
3^(T)	1.5	0.5			0.01	-0.05

Description of the results displayed on an element page

H1 Generic one element three dimensional test for arbitrary eight node element

	Coordinates			Prescribed displacement			Prescribed force
Node	X	Y	Z	u	v	w	F_x	F_y	F_z
1	-0.5	0.0	-0.5	0.015	-0.12	0.1	F_x1	F_y1	F_z1
2	1.0	0.0	0.0	u₂	0.0	0.0	0.0	F_y2	F_z2
3	1.0	1.5	0.0	0.01	0.02	0.03	F_x3	F_y3	F_z3
4	0.0	1.5	0.0	0.021	v₄	w₄	F_x4	105.0	55.0
5	0.0	0.0	1.1	0.1	0.1	0.1	F_x5	F_y5	F_z5
6	1.0	0.0	1.0	0.03	v₆	0.02	F_x6	0.0	F_z6
7	1.35	1.0	1.0	0.0	0.11	0.1	F_x7	F_y7	F_z7
8	0.0	1.0	1.0	0.03	0.02	0.03	F_x8	F_y8	F_z8

Initial configuration sensitivity used for the shape sensitivity analysis

Node	dX/dp	dY/dp
1	0	0
2	1	0
3	0	0
4	0	0
5	0	0
6	0	0
7	0	0
8	0	0

Geometric data and boundary conditions for tests of the element eigenvalues and objectivity - undistorted element shape

	Coordinates			Prescribed displacement			Prescribed force
Node	X	Y	Z	u	v	w	F_x	F_y	F_z
1	0	0	0	0.0001	-0.0002	-0.0002
2	1	0	0	0.0007	0.0001	-0.0001
3	1	1	0				0.0015	0.002	-0.001
4	0	1	0
5	0	0	1	0.00012	-0.0001	0.0002
6	1	0	1				0.0007	0.0002	0.0001
7	1	1	1				0.0015	0.002	-0.001
8^(T)	0	1	1

Geometric data and boundary conditions for tests of the element eigenvalues and objectivity - moderate distorted element shape

	Coordinates			Prescribed displacement			Prescribed force
Node	X	Y	Z	u	v	w	F_x	F_y	F_z
1	-0.5	0	-0.5	0.0001	-0.0002	-0.0002
2	1	0	0	0.0007	0.0001	-0.0001
3	1	1.5	0				0.0015	0.002	-0.001
4	0	1.5	0
5	0	0	1.1	0.00012	-0.0001	0.0002
6	1	0	1				0.0007	0.0002	0.0001
7	1.35	1	1				0.0015	0.002	-0.001
8^(T)	0	1	1

Description of the results displayed on an element page

O1 Generic one element three dimensional test for tetrahedra element

	Coordinates			Prescribed displacement			Prescribed force
Node	X	Y	Z	u	v	w	F_x	F_y	F_z
1	-0.5	0.0	-0.5	0.015	-0.12	0.1	F_x1	F_y1	F_z1
2	1.0	0.0	0.0	u₂	0.0	0.0	0.0	F_y2	F_z2
3	0.0	1.0	1.0	0.021	v₃	w₃	F_x3	10.0	5.0
4	1.0	1.5	0.0	0.01	0.02	0.03	F_x4	F_y4	F_z4

Initial configuration sensitivity used for the shape sensitivity analysis

Node	dX/dp	dY/dp
1	0	0
2	1	0
3	0	0
4	0	0

Geometric data and boundary conditions for tests of the element eigenvalues and objectivity - undistorted element shape

	Coordinates			Prescribed displacement			Prescribed force
Node	X	Y	Z	u	v	w	F_x	F_y	F_z
1	0	0	0	0.0001	-0.0002	-0.0002
2	1	0	0	0.0007	0.0001	-0.0001
3^(T)	0	0	1				0.1	-0.2	0.5
4	0	1	0

Geometric data and boundary conditions for tests of the element eigenvalues and objectivity - moderate distorted element shape

	Coordinates			Prescribed displacement			Prescribed force
Node	X	Y	Z	u	v	w	F_x	F_y	F_z
1	-0.5	0	-0.5	0.0001	-0.0002	-0.0002
2	1	0	0	0.0007	0.0001	-0.0001
3^(T)	0	0	1.1				0.1	-0.2	0.5
4	0	1.5	0

Description of the results displayed on an element page