Hierarchical higher-order dissipative methods for transient analysis

Abstract

SUMMARYThis work focuses on devising an efficient hierarchy of higher-order methods for linear transient analysis, equipped with an effective dissipative action on the spurious high modes of the response. The proposed strategy stems from the Nørsett idea and is based on a multi-stage algorithm, designed to hierarchically improve accuracy while retaining the desired dissipative behaviour. Computational efficiency is pursued by requiring that each stage should involve just one set of implicit equations of the size of the problem to be solved (as standard time integration methods) and, in addition, all the stages should share the same coefficient matrix. This target is achieved by rationally formulating the methods based on the discontinuous collocation approach. The resultant procedure is shown to be well suited for adaptive solution strategies. In particular, it embeds two natural tools to effectively control the error propagation. One estimates the local error through the next-stage solution, which is one-order more accurate, the other through the solution discontinuity at the beginning of the current time step, which is permitted by the present formulation. The performance of the procedure and the quality of the two error estimators are experimentally verified on different classes of problems. Some typical numerical tests in transient heat conduction and elasto-dynamics are presented.

Time discontinuous Galerkin methods with energy decaying correction for non‐linear elastodynamics

“…As it can be observed, the above formulation is composed of a variational statement accompanied by a discontinuous collocation of the first-order form of Equation (1) at t = t + i , with playing the role of a penalty parameter for the local initial conditions [30]. The variational statement, 330 S. DE MIRANDA, M. MANCUSO AND F. UBERTINI Equation (36), is similar to that of the classical TDG formulation, Equation (5), with the exception that the test functions belong to the space of ( p −1)-order polynomial functions on each time step.…”

Benchmark problems for wave propagation in elastic materials

“…There are no reliable numerical methods that yield an accurate solution without spurious oscillations. Current numerical approaches that treat this issue are based on the introduction of numerical dissipation or artificial viscosity from the first time increment for the suppression of spurious high-frequency oscillations; see [1][2][3][4][5][6][7][8][15][16][17][18][19][20][22][23][24][25] and others. However, numerical dissipation or artificial viscosity also affects low modes of a numerical solution.…”