No Arabic abstract
The scaled boundary finite element method (SBFEM) is capable of generating polyhedral elements with an arbitrary number of surfaces. This salient feature significantly alleviates the meshing burden being a bottleneck in the analysis pipeline in the standard finite element method (FEM). In this paper, we implement polyhedral elements based on the SBFEM into the commercial finite element software ABAQUS. To this end, user elements are provided through the user subroutine UEL. Detailed explanations regarding the data structures and implementational aspects of the procedures are given. The focus of the current implementation is on interfacial problems and therefore, element-based surfaces are created on polyhedral user elements to establish interactions. This is achieved by an overlay of standard finite elements with negligible stiffness, provided in the ABAQUS element library, with polyhedral user elements. By means of several numerical examples, the advantages of polyhedral elements regarding the treatment of non-matching interfaces and automatic mesh generation are clearly demonstrated. Thus, the performance of ABAQUS for problems involving interfaces is augmented based on the availability of polyhedral meshes. Due to the implementation of polyhedral user elements, ABAQUS can directly handle complex geometries given in the form of digital images or stereolithography (STL) files. In order to facilitate the use of the proposed approach, the code of the UEL is published open-source and can be downloaded from https://github.com/ShukaiYa/SBFEM-UEL.
The scaled boundary finite element method (SBFEM) is a semi-analytical computational scheme, which is based on the characteristics of the finite element method (FEM) and combines the advantages of the boundary element method (BEM). This paper integrates the scaled boundary finite element method (SBFEM) and the polygonal mesh technique into a new approach to solving the steady-state and transient seepage problems. The proposed method is implemented in Abaqus using a user-defined element (UEL). The detailed implementations of the procedure, defining the UEL element, updating the RHS and AMATRX, and solving the stiffness/mass matrix by the eigenvalue decomposition are presented. Several benchmark problems from seepage are solved to validate the proposed implementation. Results show that the polygonal element of PS-SBFEM has a higher accuracy rate than the standard FEM element in the same element size. For the transient problems, the results between PS-SBFEM and the FEM are in excellent agreement. Furthermore, the PS-SBFEM with quadtree meshes shows a good effect for solving complex geometric in the seepage problem. Hence, the proposed method is robust accurate for solving the steady-state and transient seepage problems. The developed UEL source code and the associated input files can be downloaded from https://github.com/yangyLab/PS-SBFEM.
This paper presents a steady-state and transient heat conduction analysis framework using the polygonal scaled boundary finite element method (PSBFEM) with polygon/quadtree meshes. The PSBFEM is implemented with commercial finite element code Abaqus by the User Element Sub-routine (UEL) feature. The detailed implementation of the framework, defining the UEL element, and solving the stiffness/mass matrix by the eigenvalue decomposition are presented. Several benchmark problems from heat conduction are solved to validate the proposed implementation. Results show that the PSBFEM is reliable and accurate for solving heat conduction problems. Not only can the proposed implementation help engineering practitioners analyze the heat conduction problem using polygonal mesh in Abaqus, but it also provides a reference for developing the UEL to solve other problems using the scaled boundary finite element method.
Consider the electromagnetic scattering of a time-harmonic plane wave by an open cavity which is embedded in a perfectly electrically conducting infinite ground plane. This paper is concerned with the numerical solutions of the transverse electric and magnetic polarizations of the open cavity scattering problems. In each polarization, the scattering problem is reduced equivalently into a boundary value problem of the two-dimensional Helmholtz equation in a bounded domain by using the transparent boundary condition (TBC). An a posteriori estimate based adaptive finite element method with the perfectly matched layer (PML) technique is developed to solve the reduced problem. The estimate takes account both of the finite element approximation error and the PML truncation error, where the latter is shown to decay exponentially with respect to the PML medium parameter and the thickness of the PML layer. Numerical experiments are presented and compared with the adaptive finite element TBC method for both polarizations to illustrate the competitive behavior of the proposed method.
In this work we formally derive and prove the correctness of the algorithms and data structures in a parallel, distributed-memory, generic finite element framework that supports h-adaptivity on computational domains represented as forest-of-trees. The framework is grounded on a rich representation of the adaptive mesh suitable for generic finite elements that is built on top of a low-level, light-weight forest-of-trees data structure handled by a specialized, highly parallel adaptive meshing engine, for which we have identified the requirements it must fulfill to be coupled into our framework. Atop this two-layered mesh representation, we build the rest of data structures required for the numerical integration and assembly of the discrete system of linear equations. We consider algorithms that are suitable for both subassembled and fully-assembled distributed data layouts of linear system matrices. The proposed framework has been implemented within the FEMPAR scientific software library, using p4est as a practical forest-of-octrees demonstrator. A strong scaling study of this implementation when applied to Poisson and Maxwell problems reveals remarkable scalability up to 32.2K CPU cores and 482.2M degrees of freedom. Besides, a comparative performance study of FEMPAR and the state-of-the-art deal.ii finite element software shows at least comparative performance, and at most factor 2-3 improvements in the h-adaptive approximation of a Poisson problem with first- and second-order Lagrangian finite elements, respectively.
In this paper, we examine the effectiveness of classic multiscale finite element method (MsFEM) (Hou and Wu, 1997; Hou et al., 1999) for mixed Dirichlet-Neumann, Robin and hemivariational inequality boundary problems. Constructing so-called boundary correctors is a common technique in existing methods to prove the convergence rate of MsFEM, while we think not reflects the essence of those problems. Instead, we focus on the first-order expansion structure. Through recently developed estimations in homogenization theory, our convergence rate is provided with milder assumptions and in neat forms.