Addressing Integration Error for Polygonal Finite Elements Through Polynomial Projections: A Patch Test Connection

Polygonal finite elements generally do not pass the patch test as a result of quadrature error in the evaluation of weak form integrals. In this work, we examine the consequences of lack of polynomial consistency and show that it can lead to a deterioration of convergence of the finite element solutions. We propose a general remedy, inspired by techniques in the recent literature of mimetic finite differences, for restoring consistency and thereby ensuring the satisfaction of the patch test and recovering optimal rates of convergence. The proposed approach, based on polynomial projections of the basis functions, allows for the use of moderate number of integration points and brings the computational cost of polygonal finite elements closer to that of the commonly used linear triangles and bilinear quadrilaterals. Numerical studies of a two-dimensional scalar diffusion problem accompany the theoretical considerations

XLME interpolants, a seamless bridge between XFEM and enriched meshless methods

In this paper, we develop a method based on local maximum entropy shape functions together with enrichment functions used in partition of unity methods to discretize problems in linear elastic fracture mechanics. We obtain improved accuracy relative to the standard extended finite element method at a comparable computational cost. In addition, we keep the advantages of the LME shape functions, such as smoothness and non-negativity. We show numerically that optimal convergence (same as in FEM) for energy norm and stress intensity factors can be obtained through the use of geometric (fixed area) enrichment with no special treatment of the nodes near the crack such as blending or shifting

Phase-field modeling of fracture in linear thin shells

We present a phase-field model for fracture in Kirchoff-Love thin shells using the local maximum-entropy (LME) meshfree method. Since the crack is a natural outcome of the analysis it does not require an explicit representation and tracking, which is advantage over techniques as the extended finite element method that requires tracking of the crack paths. The geometric description of the shell is based on statistical learning techniques that allow dealing with general point set surfaces avoiding a global parametrization, which can be applied to tackle surfaces of complex geometry and topology. We show the flexibility and robustness of the present methodology for two examples: plate in tension and a set of open connected pipes

MLS-SPH-ALE: A Review of Meshless-FV Methods and a Unifying Formulation for Particle Discretizations

Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature.[Abstract:] Mesh-based and particle methods were conceived as two different discretization strategies to solve partial differential equations. In the last two decades computational methods have diversified and a myriad of hybrid formulations that combine elements of these two approaches have been developed to solve Computational fluid dynamics problems. In this work we present a review about the meshless-FV family of methods, an analysis is carried out showing that the MLS-SPH-ALE method can be considered as a general formulation from which a set of particle-based methods can be recovered. Moreover, we show the relations between the MLS-SPH-ALE method and the finite volume method. The MLS-SPH-ALE method is a versatile particle-based method that was developed to circumvent the consistency issues of particle methods caused by the use of the kernel approximation. The MLS-SPH-ALE method is developed from the differential equation in ALE form using the partition unity property which is automatically fulfilled by the Moving Least Squares approximation.The authors gratefully acknowledge the support provided by the [Grant PID2021-125447OB-I00] funded by MCIN/AEI/ 10.13039/501100011033 and by “ERDF A way of making Europe”, and the funds by [Grant TED2021–129805B-I00] funded by MCIN/AEI/ 10.13039/501100011033 and by the “European Union NextGenerationEU/PRTR”. They also acknowledge the funding provided by the Xunta de Galicia (Grant #ED431C 2022/06). J. Fernández-Fidalgo acknowledges the support provided by “Ayudas para la recualificación del sistema universitario español para 2021–2023. Modalidad Margarita Salas RSU.UDC.MS20" by the Ministerio de Universidades of the Spanish Government and European Union through the NextGenerationEU funds.Xunta de Galicia; ED431C 2022/0

Phase-field boundary conditions for the voxel finite cell method: surface-free stress analysis of CT-based bone structures

The voxel finite cell method employs unfitted finite element meshes and voxel quadrature rules to seamlessly transfer CT data into patient-specific bone discretizations. The method, however, still requires the explicit parametrization of boundary surfaces to impose traction and displacement boundary conditions, which constitutes a potential roadblock to automation. We explore a phase-field based formulation for imposing traction and displacement constraints in a diffuse sense. Its essential component is a diffuse geometry model generated from metastable phase-field solutions of the Allen-Cahn problem that assumes the imaging data as initial condition. Phase-field approximations of the boundary and its gradient are then employed to transfer all boundary terms in the variational formulation into volumetric terms. We show that in the context of the voxel finite cell method, diffuse boundary conditions achieve the same accuracy as boundary conditions defined over explicit sharp surfaces, if the inherent length scales, i.e., the interface width of the phase-field, the voxel spacing and the mesh size, are properly related. We demonstrate the flexibility of the new method by analyzing stresses in a human femur and a vertebral body