Weighted Average Continuity Approach and Moment Correction: New Strategies for Non-consistent Mesh Projection in Structural Mechanics

Abstract

Tying non-matching meshes is needed in many instances of finite element modeling. Multiple techniques have been proposed in the literature to accomplish the correct communication between different discretizations. They all seek to achieve some trade-off in terms of accuracy, complexity and computational cost. In this work we review several of the existing techniques and benchmark them on several simple test problems in terms of accuracy and computational cost. We also discuss some of the drawbacks and limitations of the existing methods. We then propose two novel contributions. First, a new approach that imposes the continuity of the displacement field at the interface in a point-wise manner only after an integral weighted averaging procedure over each interface. Second, a procedure for the correction of the interpolation operator based on the balance of internal forces and moments at the interface is proposed, which is applicable to all the reviewed methods, both existing and the new proposed one. All the considered approaches are benchmarked on several test problems in terms of various error measures for displacements, stresses, interface forces and moments, total work at the interface and computational cost.

Appendices

Appendix 1

We provide in Fig. 25 the results of the convergence study, which led to the choice of the mesh density used in Sect. 4.1 and thereafter as reference configurations. In both configuration the reference situation chosen for this study (n=10 and m=20) show a max displacement discretization error that is less than $2\%$

Fig. 25

Convergence of the maximum displacement with mesh density : a for configuration (1) and b for configuration (2). The mesh was changed keeping the same element aspect ratio and the conforming interfaces in Fig. 13under the form ${\text{n}} \times {\text{n}} \times 10$ for the upper domain of configuration 1 and ${\text{m}} \times {\text{m}} \times 10$ for the upper domain of configuration 2

Appendix 2

In this appendix we report the detailed error analysis of each benchmark configuration, while varying the mesh density. All error measures introduced in Sect. 4.2 are provided in Figs. 26–29 for both configuration considered.

Fig. 26

Benchmark results in configuration (1) impact of n over: a$E_R$, the Resultant Force relative error, b$E_M$, the moment relative error, c$E_c$, the interface compliance relative error, d$E_d$, the displacement discontinuity relative error, e$E_{\sigma }$, the maximum of Von Mises stress relative error, f$E_U$, interface displacement field relative error, g$E_S$, average Von Mises stress relative error, h CPU time (s)

Fig. 27

Benchmark results in configuration (2)impact of m over: a$E_R$, the Resultant Force relative error, b$E_M$, the moment relative error, c$E_c$, the interface compliance relative error, d$E_d$, the displacement discontinuity relative error, e$E_{\sigma }$, the maximum of Von Mises stress relative error, f$E_U$, interface displacement field relative error, g$E_S$, average Von Mises stress relative error, h CPU time (s)

Fig. 28

Benchmark results in configuration (1) after moments balance correction. Impact of n over: a$E_R$, the Resultant Force relative error, b$E_M$, the moment relative error, c$E_c$, the interface compliance relative error, d$E_d$, the displacement discontinuity relative error, e$E_{\sigma }$, the maximum of Von Mises stress relative error, f$E_U$, interface displacement field relative error, g$E_S$, average Von Mises stress relative error, h CPU time (s)

Fig. 29

Benchmark results in configuration (2) after moments balance correction. Impact of m over: a$E_R$, the Resultant Force relative error, b$E_M$, the moment relative error, c$E_c$, the interface compliance relative error, d$E_d$, the displacement discontinuity relative error, e$E_{\sigma }$, the maximum of Von Mises stress relative error, f$E_U$, interface displacement field relative error, g$E_S$, average Von Mises stress relative error, h CPU time (s)