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We present a new accuracy condition for the construction of continuous mass-lumped elements. This condition is less restrictive than the one currently used and enabled us to construct new mass-lumped tetrahedral elements of degrees 2 to 4. The new degree-2 and degree-3 tetrahedral elements require 15 and 32 nodes per element, respectively, while currently, these elements require 23 and 50 nodes, respectively. The new degree-4 elements require 60, 61, or 65 nodes per element. Tetrahedral elements of this degree had not been found until now. We prove that our accuracy condition results in a mass-lumped finite element method that converges with optimal order in the $L^2$-norm and energy-norm. A dispersion analysis and several numerical tests confirm that our elements maintain the optimal order of accuracy and show that the new mass-lumped tetrahedral elements are more efficient than the current ones Read More: https://epubs.siam.org/doi/abs/10.1137/18M1175549?af=R&mobileUi=0&

This paper describes a a numerical algorithm to improve the poor performance of a four-node linear tetrahedral element. For this purpose a novel finite volume method utilizing an arbitrary median-meshed polyhedral (AMP) cell is developed. The AMP cells are constructed uniquely by connecting the median points of unstructured tetrahedra: the edge mid-points; the face centers; the volume centroids. These cells are used as control volumes to descretize the equations of the conservation laws by the finite volume method and to solve constitutive equations. The derivation of the equation of motion to accelerate computational nodes is detailed. The present method is implemented in a general-purpose explicit Lagrangian program, which is applied to solve Taylor impact problems in order to evaluate the performance of the AMP cells. Test cases of the three different materials as Aluminum, Copper and Steel are selected. The deformed shapes of the test pieces obtained by the analyses show good agreements with the ones by the experiments even if the computational nodes are distributed nonuniformly. Computational results are also compared about plastic strains obtained by the eight-node hexahedral elements of the same computer program.

In this work finite element simulations are conducted based on the micro structure of polymers in order to transfer the information of the micro level to the macro level. The micro structure of polymers is characterized by chain-like macromolecules linked together at certain points. In this way an irregular three-dimensional network is formed. Many authors use the tool of statistical mechanics to describe the deformation behaviour of the entire network. Most of these concepts can be reformulated as traditional continuum mechanical formulations. They are, however, restricted to affine deformation, regular chain arrangements and purely elastic material behaviour. For this reason, in the present contribution, we propose a new finite element-based simulation method for polymer networks which enables us to include non-affinity and arbitrary chain configurations. It can be easily extended to include chain breakage and reconnection. The polymer structure to be investigated, e.g. a rubber boot or a seal, is discretized by means of tetrahedral elements. To each edge of a tetrahedral element one truss element is attached which models the force–stretch behaviour of a bundle of polymer chains. Each of these tetrahedral unit cells represents the micro mechanical material behaviour in a certain point of the network. The proposed method provides the possibility to observe how changes at the microscopic level influence the macroscopic material behaviour. Such information is especially valuable for the polymer industry.

A framework for the construction of stable spectral methods on arbitrary domains with unstructured grids is presented. Although most of the developments are of a general nature, an emphasis is placed on schemes for the solution of partial differential equations defined on the tetrahedron. In the first part the question of well-behaved multivariate polynomial interpolation on the tetrahedron is addressed, and it is shown how to extend the electrostatic analogy of the Jacobi polynomials to problems beyond the line. This allows for the identification of nodal sets suitable for polynomial interpolation within the tetrahedron and, subsequently, for the formulation of stable spectral schemes on such unstructured nodal sets. The second part of this work is devoted to a discussion of weakly imposed boundary conditions, and energy-stable schemes are formulated for a wide class of problems, exemplified by advection problems, advection-diffusion problems, and linear symmetric hyperbolic systems. Finally, in the third part, issues related to computational efficiency and implementation of the schemes are discussed. The spectral accuracy of the approximation is confirmed through an example, and factorization methods for the efficient computation of derivatives on the general nodal sets within the d-simplex are developed, ensuring that the proposed schemes are competitive with tensor-product-based methods. In this last part we also show that the advective operator results in an ${\cal O}(n^{-2})$ restriction on the time-step, similar to that of spectral collocation methods employing a tensor-product-based approximation. The performance of the proposed scheme is illustrated by solving a wave problem on a triangulated domain, confirming the expected accuracy and stability.