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In this contribution, the finite cell method (FCM) is applied to solve transient problems of linear elastodynamics. The mathematical formulation of FCM for linear elastodynamics is presented, following from the weak formulation of the initial/boundary-value problem. Semi-discrete time integration schemes are briefly discussed, and the choice of implicit time integration is justified. A 1D benchmark problem is solved using FCM, illustrating the method's ability to solve problems of linear elastodynamics obtaining high rates of convergence. Furthermore, a numerical example of transient analysis from an industrial application is solved using FCM. The numerical results are compared to the results obtained using state-of-the-art commercial software, employing linear finite elements, in conjunction with explicit time integration. The results illustrate the potential of FCM as a powerful tool for transient analysis in elastodynamics, offering a high degree of accuracy at a moderate computational effort.

Metal lattice structures filled with a damping material such as polymer can exhibit high stiffness and good damping properties. Mechanical simulations of parts made from these composites can however require a large modeling and computational effort because relevant features such as complex geometries need to be represented on multiple scales. The finite cell method (FCM) and numerical homogenization are potential remedies for this problem. Moreover, if the microstructures are placed in between the components of assemblies for vibration reduction, a modified mortar technique can further increase the efficiency of the complete simulation process. With this method, it is possible to discretize the components separately and to integrate the viscoelastic behavior of the composite damping layer into their weak coupling. This paper provides a multiscale computational material design framework for such layers, based on FCM and the modified mortar technique. Its efficiency even in the case of complex microstructures is demonstrated in numerical studies. Therein, computational homogenization is first performed on various microstructures before the resulting effective material parameters are used in larger-scale simulation models to investigate their effect and to verify the employed methods.

In many extended versions of the finite element method (FEM) the mesh does not conform to the physical domain. Therefore, discontinuity of variables is expected when some elements are cut by the boundary. Thus, the integrands are not continuous over the whole integration domain. Apparently, none of the well developed integration schemes such as Gauss quadrature can be used readily. This paper investigates several modifications of the Gauss quadrature to capture the discontinuity within an element and to perform a more precise integration. The extended method used here is the finite cell method (FCM), an extension of a high-order approximation space with the aim of simple meshing. Several examples are included to evaluate different modifications.

In this paper, a new boundary-conforming adaptive method for the numerical integration of trimmed elements is presented. The locations and weights of new integration points are determined based on special mapping formulations. The prerequisite of this technique is to describe the trimming curves/surfaces by parametric Bezier curves/surfaces within the parent element domain. The fitting error is under control, therefore the number of quadrature points for exact integration can be adjusted automatically based on the complexity of trimming curve and the corresponding integrand. The proposed method can be easily implemented into fictitious domain approaches. Different integration tasks as well as structural examples reveal that the proposed method delivers accurate and robust solutions for a wide variety of 2D/3D geometries with a rather low number of integration points.