An Adaptive Polygonal Scaled Boundary Finite Element Method for Elastodynamics

2016 ◽  
Vol 13 (02) ◽  
pp. 1640015 ◽  
Author(s):  
Z. H. Zhang ◽  
Z. J. Yang ◽  
J. H. Li
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Dynamic Responses ◽  
Adaptive Finite Element Method ◽  
Error Estimator ◽  
State Variables ◽  
Time Step ◽  
Polygonal Mesh ◽  
Scaled Boundary Finite Element ◽  
Element Method

An adaptive polygonal scaled boundary finite element method (APSBFEM) is developed for elastodynamics. Flexible polygonal meshes are generated from background Delaunay triangular meshes and used to calculate structure’s dynamic responses. In each time step, a posteriori-type energy error estimator is employed to locate the polygonal subdomains with exceeding spatial discretization error, then edge midpoints of the corresponding triangles are inserted into the background. A new Delaunay triangular mesh and a polygonal mesh are regenerated successively. The state variables, including displacement, velocity and acceleration are mapped from the old polygonal mesh to the new one by a simple algorithm. A benchmark elastodynamic problem is modeled to validate the developed method. The results show that the adaptive meshes are capable of capturing the steep stress regions, and the dynamic responses agree well with those from the adaptive finite element method and the polygonal scaled boundary finite element method without adaptivity using fine meshes.

scholarly journals Quasi-optimality of an Adaptive Finite Element Method for Cathodic Protection

2019 ◽  
Vol 53 (5) ◽  
pp. 1645-1665
Author(s):  
Guanglian Li ◽  
Yifeng Xu
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Cathodic Protection ◽  
Finite Element Approximation ◽  
Adaptive Finite Element Method ◽  
Error Estimator ◽  
Element Approximation ◽  
Adaptive Finite Element ◽  
Energy Error ◽  
Element Method

In this work, we derive a reliable and efficient residual-typed error estimator for the finite element approximation of a 2D cathodic protection problem governed by a steady-state diffusion equation with a nonlinear boundary condition. We propose a standard adaptive finite element method involving the Dörfler marking and a minimal refinement without the interior node property. Furthermore, we establish the contraction property of this adaptive algorithm in terms of the sum of the energy error and the scaled estimator. This essentially allows for a quasi-optimal convergence rate in terms of the number of elements over the underlying triangulation. Numerical experiments are provided to confirm this quasi-optimality.

scholarly journals A Polygonal Scaled Boundary Finite Element Method for Solving Heat Conduction Problems

2021 ◽  
Author(s):  
Yang Yang ◽  
Zongliang Zhang ◽  
Yelin Feng ◽  
Yuzhen Yu ◽  
Kun Wang ◽  
...  
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Heat Conduction ◽  
Heat Conduction Problem ◽  
Mass Matrix ◽  
Eigenvalue Decomposition ◽  
Benchmark Problems ◽  
Polygonal Mesh ◽  
Scaled Boundary Finite Element ◽  
Element Method

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 Subroutine (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.

scholarly journals Free and Forced Vibration Analysis in Abaqus Based on the Polygonal Scaled Boundary Finite Element Method

2021 ◽  
Vol 2021 ◽  
pp. 1-17
Author(s):  
Nan Ye ◽  
Chao Su ◽  
Yang Yang
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Forced Vibration ◽  
Eigenvalue Decomposition ◽  
Benchmark Problems ◽  
Polygonal Mesh ◽  
Finite Element Software ◽  
Forced Vibration Analysis ◽  
Scaled Boundary Finite Element ◽  
Element Method

The polygonal scaled boundary finite element method (PSBFEM) is a novel approach integrating the standard scaled boundary finite element method and the polygonal mesh technique. In this work, a user-defined element (UEL) for dynamic analysis based on the PSBFEM is developed in the general finite element software ABAQUS. We present the main procedures of interacting with Abaqus, updating AMATRX and RHS, defining the UEL element, and solving the stiffness and mass matrices through eigenvalue decomposition. Several benchmark problems of free and forced vibration are solved to validate the proposed implementation. The results show that the PSBFEM is more accurate than the FEM with mesh refinement. Moreover, the PSBFEM avoids the occurrence of hanging nodes by constructing a polygonal mesh. Thus, the PSBFEM can choose an appropriate mesh resolution for different structures ensuring accuracy and reducing calculation costs.

AN ADAPTIVE TIME-STEPPING PROCEDURE BASED ON THE SCALED BOUNDARY FINITE ELEMENT METHOD FOR ELASTODYNAMICS

2012 ◽  
Vol 09 (01) ◽  
pp. 1240007 ◽  
Author(s):  
Z. H. ZHANG ◽  
Z. J. YANG ◽  
GUOHUA LIU
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Local Error ◽  
Error Estimator ◽  
Time Step ◽  
Time Stepping ◽  
Time Increment ◽  
Adaptive Time ◽  
Adaptive Time Stepping ◽  
Scaled Boundary Finite Element

This study develops an adaptive time-stepping procedure of Newmark integration scheme for transient elastodynamic problems, based on the semi-analytical scaled boundary finite element method (SBFEM). In each time step, a posteriori local error estimator based on the linear distributed acceleration is employed to estimate the error caused by the time discretization. The total energy of the domain, consisting of the kinetic energy and the strain energy, is calculated semi-analytically. The time increment is automatically adjusted according to a simple criterion. Three examples with stress wave propagation were modeled. The numerical results show that the developed method is capable of limiting the local error estimator within specified targets by using an optimal time increment in each time step.

An Optimal Adaptive Finite Element Method for an Obstacle Problem

2015 ◽  
Vol 15 (3) ◽  
pp. 259-277 ◽  
Author(s):  
Carsten Carstensen ◽  
Jun Hu
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Obstacle Problem ◽  
Posteriori Error ◽  
Adaptive Finite Element Method ◽  
Error Estimator ◽  
A Posteriori ◽  
Optimal Complexity ◽  
A Posteriori Error ◽  
Element Method

AbstractThis paper provides a refined a posteriori error control for the obstacle problem with an affine obstacle which allows for a proof of optimal complexity of an adaptive algorithm. This is the first adaptive mesh-refining finite element method known to be of optimal complexity for some variational inequality. The result holds for first-order conforming finite element methods in any spacial dimension based on shape-regular triangulation into simplices for an affine obstacle. The key contribution is the discrete reliability of the a posteriori error estimator from [Numer. Math. 107 (2007), 455–471] in an edge-oriented modification which circumvents the difficulties caused by the non-existence of a positive second-order approximation [Math. Comp. 71 (2002), 1405–1419].

3D direct current resistivity modeling with unstructured mesh by adaptive finite-element method

Geophysics ◽  
2010 ◽  
Vol 75 (1) ◽  
pp. H7-H17 ◽  
Author(s):  
Zhengyong Ren ◽  
Jingtian Tang
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Direct Current ◽  
Homogeneous Model ◽  
Posteriori Error ◽  
Adaptive Finite Element Method ◽  
Error Estimator ◽  
Adaptive Finite Element ◽  
Resistivity Modeling ◽  
Element Method

A new adaptive finite-element method for solving 3D direct-current resistivity modeling problems is presented. The method begins with an initial coarse mesh, which is then adaptively refined wherever a gradient-recovery-based a posteriori error estimator indicates that refinement is necessary. Then the problem is solved again on the new grid. The alternating solution and refinement steps continue until a given error criterion is satisfied. The method is demonstrated on two synthetic resistivity models with known analytical solutions, so the errors can be quantified. The applicability of the numerical method is illustrated on a 2D homogeneous model with a topographic valley. Numerical results show that this method is efficient and accurate for geometrically complex situations.

scholarly journals An Adaptive Finite Element Method for Stationary Incompressible Thermal Flow Based on Projection Error Estimation

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
Zhili Guo ◽  
Jian Su ◽  
Hao Chen ◽  
Xiaomin Liu
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Stationary Flow ◽  
Adaptive Strategy ◽  
Posteriori Error ◽  
Adaptive Finite Element Method ◽  
Error Estimator ◽  
Adaptive Finite Element ◽  
Thermal Flow ◽  
Element Method

An adaptive finite element method is presented for the stationary incompressible thermal flow problems. A reliable a posteriori error estimator based on a projection operator is proposed and it can be computed easily and implemented in parallel. Finally, three numerical examples are given to illustrate the efficiency of the adaptive finite element method. We also show that the adaptive strategy is effective to detect local singularities in the physical model of square cavity stationary flow in the third example.

scholarly journals PSBFEM-Abaqus: Development of User Element Subroutine (UEL) for Polygonal Scaled Boundary Finite Element Method in Abaqus

2021 ◽  
Vol 2021 ◽  
pp. 1-22
Author(s):  
Nan Ye ◽  
Chao Su ◽  
Yang Yang
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Eigenvalue Decomposition ◽  
Benchmark Problems ◽  
Polygonal Mesh ◽  
Finite Element Software ◽  
Linear Elastostatics ◽  
Novel Method ◽  
Scaled Boundary Finite Element ◽  
Element Method

The polygonal scaled boundary finite element method (PSBFEM) is a novel method integrating the standard scaled boundary finite element method (SBFEM) and the polygonal mesh technique. This work discusses developing a PSBFEM framework within the commercial finite element software Abaqus. The PSBFEM is implemented by the User Element Subroutine (UEL) feature of the software. The details on the main procedures to interact with Abaqus, defining the UEL element, and solving the stiffness matrix by the eigenvalue decomposition are present. Moreover, we also develop the preprocessing module and the postprocessing module using the Python script to generate meshes automatically and visualize results. Several benchmark problems from two-dimensional linear elastostatics are solved to validate the proposed implementation. The results show that PSBFEM-UEL has significantly better than FEM convergence and accuracy rate with mesh refinement. The implementation of PSBFEM-UEL can conveniently use arbitrary polygon elements by the polygon/quadtree discretizations in the Abaqus. The developed UEL and the associated input files can be downloaded from https://github.com/hhupde/PSBFEM-Abaqus.

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