VARIATIONAL MASS SCALING FOR ISOGEOMETRIC FINITE ELEMENTS

Abstract eng:
The primal goal of this contribution is the investigation of recently developed variational selective mass scaling techniques [3] for isogeometric finite elements. They allow an increase of the critical time step size for explicit time integration without substantial loss in accuracy in the lower modes. Especially for low order FEM, scaling of inertia for explicit time integration is a common procedure. Conventional mass scaling (CMS), which preserves the diagonal form of the lumped mass matrix and adds artificial mass only to the diagonal, was recently also applied to isogeometric shell elements [1]. The drawback of the method is that the translational and rotational inertia of the structure increases, which may cause non-physical phenomena. Algebraic selective mass scaling (ASMS) adds artificial terms both to the diagonal and non-diagonal terms, which results in a non-diagonal mass matrix, but at least may allow preservation of translational inertia or may allow higher order accurate mass matrices [2]. Both CMS and ASMS lack a rigorous variational formulation and only the former preserves the diagonal structure of the mass. Variationally based selective mass scaling (VSMS) for a wide range of elements and reciprocal mass matrices (VSRMM) for simplex elements were recently proposed in [3] and [4], respectively. In this contribution, both methods are applied to 1D-B-Spline and 2D- and 3D-NURBS finite elements. Starting point of both methods is a parametrized variational principle of elastodynamics, which can be interpreted as a penalized Hamilton’s principle which imposes relations between the independent fields. A consistent discretization of the principle results in parametric families of mass matrices. For VSMS, a mathematically consistent, but non-diagonal mass matrix is obtained, therefore the solution of a linear system of equations at each time step is required. For VSRMM, a sparse inverse of the mass matrix is obtained directly, thus avoiding the additional expense of equation solving. For the efficiency and accuracy of VSMS and VSRMM, the choice of ansatz spaces is crucial. A powerful tool for this purpose is the dispersion analysis. It allows to compare different choices of ansatz spaces with regard to their influence on the maximum eigenfrequency and thus on the maximal possible time step size and the accuracy (order of convergence). A recommendation for the best choice of ansatz functions is given and the efficiency of the proposed approach is investigated by numerical examples.

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