Efficiency test

Measurement of the numerical efficiency of the element is based on the measurement of the actual evaluation time. The 1×1 square or 1×1×1 cube, supported as depicted below, is taken as a standardized example. The uniform load is applied on the top surface. Resulting uniform strain and stress field inside the body ensures that all material points are in the same physical state (e.g. elastic or plastic).

The efficiency of the element formulation is measured at the load level that is from the computational point most unfavourable. Results obtained for the most unfavourable load level are displayed on the element's home page as follows

Floating point operations per element tangent and residual: N_flop
Assembly time: t_a Linear solution time: t_s Threshold: T

where N_flop is the equivalent number of floating point operations needed for the evaluation of the element tangent matrix and residual, t_a is the total evaluation time in seconds used for the assembly of the global tangent matrix and residual, t_s is the total evaluation time in seconds used for the solution of the resulting linear system of equations, and threshold T is the number of elements when the linear solution time equals the assembly time (t_s(N)=t_a(N)). Let N be the number of elements and Flops the number of floating point operations per seconds on the machine where the test is performed then N_flop is expressed by

and threshold T for the regular 2D and 3D meshes depicted below by

.

It can be shown that so defined threshold value does not depend on the actual number of elements but only on the type of elements and the general topology of the problem. For the regular 2D mesh and for the standard profile solver the evaluation of the threshold value T leads to

where n is the number of subdivisions of the regular square or cube in each spatial direction, n_eq is the number of equations, f_d.o.f is the number of degrees of freedom per nodal point, bw is the average band width of the resulting linear system of equations, and f_s , f_bw , f_a, f_ne are the proportionality factors that depend on the topology of the problem, and the type of the element. The same result can be obtained for the regular 3D mesh.

Typical meshes used for the estimation of the element numerical efficiency are depicted below.

T1,T2 triangular mesh

T1: Nodes: 5041 Elements: 9800 T2: Nodes: 5041 Elements: 2450

Q1, Q2 quadrilateral mesh

Q1: Nodes: 5041 Elements: 4900 Q2: Nodes: 5041 Elements: 1225

O1 tetrahedral mesh

O1: Nodes: 1331 Elements: 5200

H1 hexahedral mesh

H1: Nodes: 3375 Elements: 2744