Advanced tetrahedra

Completed Research project

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

Tetrahedral finite element, as it is used for automatic meshing for complex geometries.


  • Reciprocal mass matrices for tetrahedral finite elements with Allman's rotations
  • Advanced nodal time step estimates
  • Variational strain scaling for composite tetrahedral finite elements with Allman's 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.

Tetrahedral 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 hexahedral 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
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

Finite elements with Allman's rotations provide good computational efficiency for solid problems and explicit codes exhibiting less locking than linear elements and lower computational cost than quadratic finite elements. These elements possess a compatible interpolation of the displacements with only vertex degrees of freedom. Taking a standard quadratic element as a base, the midside nodal displacements are then constrained to vertex displacements and rotations of the edge by a transformation matrix. A significant increase in the critical time step can be achieved by a specially constructed reciprocal mass matrix. The reciprocal mass matrix allows the direct computation of the acceleration vector from the total force vector by multiplication with a sparse matrix. Therefore, each time step is similar expensive as for standard diagonal mass matrices. Additionally, the time step can be increased by a factor of from 8% to 100%. The construction used herein combines a variational construction for vertex displacements with diagonal inertia for Allman’s rotations.

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

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

Advanced nodal time step estimates

Elements with Allman’s rotations present a challenge for an affordable and accurate estimate of the feasible time step. The presence of mixed physical units for displacements (in m) and rotations (no unit) destroys tightness of standard nodal estimates based on the Gershgorin circle theorem. The advanced estimate combines similarity transformation to correct the mismatch of the units and more accurate eigenvalue bound using Ostrowski’s circle theorem.

The figure below shows the comparison of the standard nodal time estimate and the proposed one for NAFEMS benchmark FV32. The proposed estimate leaves a much smaller gap (14.2%) to the exact feasible time step than standard Gershgorin (67.0%) and a combination of Gershgorin with similarity transformation (16.4%).

Advanced nodal time step estimates
5th eigenmode and comparison of tightness for time step estimates for a triangular element with Allman’ rotations (Tri3R)

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.

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.


  1. Tkachuk, A. (2020). Customization of reciprocal mass matrices via log-det heuristic. International Journal for Numerical Methods in Engineering, 121, 690–711.
  2. Tkachuk, A. (2020). Reciprocal mass matrices and a feasible time step estimator for finite elements with Allman’s rotations. International Journal for Numerical Methods in Engineering, 122, 1401–1422.
  3. Tkachuk, A., Kolman, R., Gonzalez, J. A., Bischoff, M., & Kopacka, J. (2019). Time step estimates for reciprocal mass matrices using Ostrowski’s bounds. Proc. 7th ECCOMAS Thematic Conference on Computational Methods in Structural Dynamics and Earthquake Engineering, M. Papadrakakis and M. Fragiadakakis (eds.), Crete, Greece, June 24-26. 2019.

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
Anton Tkachuk


Research partners

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