High‐order gaussian quadrature in X‐FEM with the lagrange‐multiplier for fluid–structure coupling

Abstract

SUMMARYThis paper presents an alternative choice for the construction of integration cells and an additional Lagrange-multiplier mesh in computations using the eXtended finite element method (X-FEM) with the Lagrange-multiplier. The proposed computational target is flow simulations with the Dirichlet boundary conditions on a non-boundary-fitted mesh. The methods studied in this paper make use of a high-order Gaussian quadrature to relax the complexity in implementing the X-FEM with the Lagrange-multiplier. The first of these is the straightforward employment of the Gaussian quadrature as a numerical integration scheme for enriched fluid elements. The second makes use of free Lagrange-multipliers that are not placed at crossing points between fluid elements and the Dirichlet boundary. This method considers the scalability of fluid-structure and fluid-thin object interactions. A unique benchmark problem in which a computational fluid domain is partitioned into two physical domains with the Lagrange-multipliers is introduced to validate the proposed method. Numerical results show that this method is applicable in practice with acceptable numerical accuracy.

LLM and X-FEM based interface modeling of fluid–thin structure interactions on a non-interface-fitted mesh

“…These basic components have already been proposed in Refs. [34,60] with detailed examinations of the integration accuracy and applicability of the high-order Gaussian quadrature where a flow problem with a domain-partitioning boundary is used for the examinations [60]. This paper shows the FSI extension and application and discusses how the interface discontinuity enhances the simulation results, and how omitting the discontinuity affects them.…”

“…In particular, recent interest in the non-interface-fitted mesh approaches [1,10,11,13,[17][18][19]21,22,24,25,[27][28][29][32][33][34][35][36][37]40,41,[46][47][48][54][55][56][58][59][60][61][62], in which the fluid and structure are described in general by the Eulerian and Lagrangian coordinates, respectively, is remarkable. One reason for this interest is associated with the practical merit of mesh generation, because the approach using a non-interface-fitted mesh can reduce human and computational costs and difficulties of the mesh generation.…”

“…This set of functions is considered suitable for the fluid-shell coupled simulations where the neutral plane of the shell element is the computational interface. The third feature is to apply high-order Gaussian quadrature to domain integrals of the enriched fluid elements and surface integrals of the Lagrange-multipliers [60], which removes fully the subdivisions from the FSI computation scheme which uses the Lagrange-multiplier method and the X-FEM. These basic components have already been proposed in Refs.…”

“…This approach is able to simplify the coupling scheme, because the intersection-based subdivisions [27,47,48,54] of the interface are not necessary throughout the simulation. The second feature is to employ the edge and Heaviside functions as the respective enrichment functions of the interface velocity and pressure [28,34,60]. This set of functions is considered suitable for the fluid-shell coupled simulations where the neutral plane of the shell element is the computational interface.…”

“…The first feature of the present method is to discretize the Lagrange-multiplier at the Lagrangian nodes of the structure element [34,60]. This approach is able to simplify the coupling scheme, because the intersection-based subdivisions [27,47,48,54] of the interface are not necessary throughout the simulation.…”