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From a computational perspective, the analysis of wave propagation and other high-frequency dynamic simulations is rather costly and presents considerable challenges. In such cases, explicit time integration is frequently employed, as the time step for numerical simulation is limited by the physical characteristics of the system. Nevertheless, the efficiency of explicit time integration depends on the availability of a lumped mass matrix (LMM). As a matter of fact, no mass lumping technique is able to deliver satisfying accuracy for all types of elements. However, the spectral element method (SEM) is widely utilized for transient simulations as it provides a lumped mass matrix (LMM) by construction and provides optimal rates of convergence for quadrilateral and hexahedral elements. In this context, lumping is achieved through the application of the nodal quadrature technique, where Gauß-Lobatto-Legendre (GLL) points are utilized both for the definition of Lagrangian shape functions and the quadrature rule. It should be noted, however, that this method is not suitable for other point distributions, as it can result in zero or negative masses, which makes it inappropriate for arbitrary element types. Conversely, established mass lumping techniques, such as the row-sum method and the diagonal scaling method (HRZ lumping), can be employed to transform a consistent mass matrix into a diagonal matrix for arbitrary element types. These approaches frequently cause complications such as reduced convergence rates and an inability to guarantee the positive definiteness of the mass matrix. Therefore, it is highly desirable to develop a new method that addresses these shortcomings.
A first step in this direction is achieved in the contribution at hand. Here, we introduce a novel approach based on the spectral decomposition theorem (SDT) that enables the transformation of a consistent mass matrix (cSEM) into a lumped mass matrix (lSEM). This technique ensures that the lumped mass matrix is positive definite and guarantees exponential convergence rates, as it exactly reproduces the SEM mass matrix obtained through nodal quadrature. The proposed SDT-based mass lumping method thus provides a foundation for the development of advanced mass lumping schemes for other element types, for which no such methods currently exist.
