Novel Mass Lumping Approach Leveraging the Spectral Decomposition Theorem

simulations are very time-consuming and challenging to deal with. For such problems, explicit time integration is often employed, since the time step size for the numerical simulation is already limited by the physics of the problem. However, for explicit time integration methods to be competitive, a lumped mass matrix (LMM) must be available. Unfortunately, there is no mass lumping approach that provides satisfying accuracy for all element types. The spectral element method (SEM) is a frequently utilized methodology for obtaining an LMM, as it allows an optimal convergence rate for rectangular and hexahedral Lagrange elements and thus fulfills the combination of diagonal structure and accuracy in the best possible way to date. In this case, lumping is achieved by using the nodal quadrature technique, where Gauß-Lobatto-Legendre (GLL) points are utilized both for defining the Lagrangian shape functions and the quadrature rule. This is not a recommended approach for other point distributions, as it is possible for zero or negative masses to occur, and is therefore inappropriate for arbitrary element types. On the other hand, there are the well-known mass lumping methods, e.g., the row-sum technique and the diagonal scaling method (HRZ lumping) which can be employed for arbitrary elements to transform a consistent mass matrix in a diagonal structure. Unfortunately, these methods cause problems such as deteriorated convergence rates and do not ensure the positive-definiteness of the mass matrix. A new method that can free itself from these shortcomings would therefore be ideal. In this article a novel approach based on the spectral decomposition theorem (SDT) is presented that allows to transform a consistent SEM mass matrix (cSEM) into a lumped SEM mass matrix (lSEM). This technique guarantees positive-definiteness of the lumped mass matrix and exponential convergence rates, since it reproduces the SEM mass matrix obtained by nodal quadrature exactly. The proposed SDT mass lumping approach thus provides a strong foundation for developing advanced mass lumping schemes for other element types, for which no such methods currently exist.