DOI：10.1007/s00466-024-02501-4

Abstract：Abstract In this work, we present a frst-order stabilization-free virtual elementmethod (SFVEM) for three-dimensional hyperelastic problems. Different from theconventional virtual element method, which necessitates additional stabilizationterms in the bilinear formulation, the method developed in this work operates withoutthe need for any stabilization. Consequently, it proves highly suitable for the compu-tation of nonlinear problems. Similar to the techniques used in the two-dimensionastabilization-free virtual element method, the new virtual element space is modifiedto allow the computation of the higher-order L2 projection of the gradient. This pa-per reviews the calculation process of the traditional H projection operator, anddescribes in detail how to calculate the high-order Lz projection operator for three.dimensional problems. Based on this high-order Lz projection operator, this paperextends the method to more complex three-dimensional nonlinear problems. Somebenchmark problems illustrate the capability of the stabilization-free VEM for three-dimensional hyperelastic problems.