Research project

Advanced tetrahedra

Methods for selective scaling of strain and inertia for explicit dynamics with tetrahedral finite elements.

Tetrahedron (c)
Tetrahedral finite element, as it is used for automatic meshing for complex geometries.


  • Variational inertia scaling for tetrahedral finite elements with Allman rotations
  • Variational inertia scaling for composite tetrahedral finite elements
  • Variational strain scaling for composite tetrahedral finite elements with Allman rotations
  • Reduction of the numerical dispersion and reflection transmission error

Project description

The purpose of the proposed research project is the development of variational methods for selective scaling of inertia and strain for explicit dynamic finite element analysis to develop efficient and accurate tetrahedral finite elements for non-linear dynamics and wave propagation.

Tetrahedal vs. hexahedral finite elements

For complex geometries, tetrahedral finite elements are often indispensable in finite element meshes. Although the model of a tool wrench was greatly simplified in Figure 1, it can only be automatically meshed with tetrahedral finite elements. However, existing tetrahedral finite elements are limited in their performance due to the following issues: Locking, high CPU cost, high numerical dispersion and large reflection-transmission errors in heterogeneous media. So far, the universal accurate high-performance tetrahedral does not exist. The simplified model of the tool wrench shows that for a given element edge length, six times more degrees of freedom are required for the tetrahedral mesh than for the hexaheral one, resulting in a higher calculation effort. The effort is additionally increased by the critical time step, which is smaller by a factor of 1.7 for the tetrahedral finite element mesh. Despite the higher computational effort, the displacement solution for the tetrahedral elements is clearly too conservative due to artificial stiffening effects.

four- and three-noded finite elements (c)
Comparison of four- and three-noded finite elements in terms of efficiency and accuracy

The focus of this research project is to reduce locking effects and CPU cost, since these problems are particularily important in non-linear structural dynamics. The variationally selective inertia and strain scaling is investigated and transferred to common tetrahedron finite elements.

Inertia scaling

Preliminary work has shown that a significant increase of the critical time step can be achieved by variational inertia scaling. Variationally scaled mass matrices as well as reciprocal mass matrices have been proposed. Whereas the variationally scaled mass matrices are not diagonal and therefore require the solution of a linear system of equations in each time step, the reciprocal mass matrices allow a trivial solution. Each time step is therefore similar expensive as for the usually used diagonal mass matrix. Additionally, the time step can be increased by a factor of approximately 2.

Applying the idea of inertia scaling to numerically expensive but accurate element formulations such as elements with Allman rotations and composite elements leads to accurate and efficient tetrahedral elements that perform similarly well than hexahedral finite elements.

The figure below shows the analysis of a cantilever beam under an abruptly applied constant force F. The application of inertia scaling to the four-noded tetrahedral finite element with Allman rotations allows a reduction of the calculation effort by 53%. In the result for the deflection, no difference to the solution with consistent or diagonal mass matrix can be determined.

Inertia scaling tetrahedral Allman rotations (c)
Inertia scaling for a tetrahedral element with Allman rotations (TET4R)

Strain scaling

For elements that are efficient (or whose efficiency has been improved by inertial scaling) but show artificial locking effects, variationally consistent strain scaling can be applied.  As an example, an optimized dispersion behavior for tetrahedral elements of higher order can be achieved by specific adjustment of the strain scaling for wave propagation problems.

Heterogeneous material

In preliminary studies it was observed that even though very good results were achieved for homogeneous materials with the variationally scaled reciprocal mass, only unsatisfactory results were achieved for heterogeneous materials. An improved construction of the biorthogonal ansatz spaces of the multi-field formulation has led to significantly improved convergence properties, see figure below. The analysis of the reflection transmission error confirms this result.

convergence (c)
Improved convergence behavior for a member modeled from two materials

Templates based on variationally parameterized principles

Variational inertia scaling and strain scaling can be developed as templates based on variationally parameterized principles. The idea of mass and stiffness templates was proposed in 1994 by Prof. Carlos Felippa from Colorado University in Boulder. The templates based on variationally parameterized principles allow a general construction of parameterized mixed variational principles. By selecting appropriate free parameters and ansatz spaces for the independent variables, families of new finite elements with improved properties are proposed.

Project data

Project titel:
Methods for selective scaling of strain and inertia for explicit dynamics with tetrahedral finite elements
German Research Foundation (DFG), Research Grant TK 63/1-1, GEPRIS project number 326748051


Research partners


Anton Tkachuk

Anton Tkachuk


To the top of the page