A three-dimensional adaptive analysis using the meshfree node-based smoothed point interpolation method (NS-PIM)

In this paper, a three-dimensional (3-D) adaptive analysis procedure is proposed using the meshfree node-based smoothed point interpolation method (NS-PIM). Previous study has shown that the NS-PIM works well with the simplest four-node tetrahedral mesh, which is easy to be implemented for complicated geometry. In contrast to the displacement-based FEM providing lower bound solutions, the NS-PIM possesses the attractive property of providing upper bound solutions in strain energy norm. In the present adaptive procedure, a novel error indicator is devised for NS-PIM settings, which evaluates the maximum difference of strain energy values among four nodes in each of the tetrahedral cells. A simple h-type local refinement scheme is adopted and coupled with 3-D mesh automatic generator based on Delaunay technology. Numerical results indicate that the adaptive refinement procedure can effectively capture the stress concentration and solution singularities, and carry out local refinement automatically. The present adaptive procedure achieves much higher convergence in strain energy solution compared to the uniform refinement, and obtains upper bound solution in strain energy efficiently for force driven problems.     

An automatic adaptive refinement procedure for the reproducing kernel particle method. Part II: Adaptive refinement

In Part II of this study, an automatic adaptive refinement procedure using the reproducing kernel particle method (RKPM) for the solution of 2D linear boundary value problems is suggested. Based in the theoretical development and the numerical experiments done in Part I of this study, the Zienkiewicz and Zhu (Z–Z) error estimation scheme is combined with a new stress recovery procedure for the a posteriori error estimation of the adaptive refinement procedure. By considering the a priori convergence rate of the RKPM and the estimated error norm, an adaptive refinement strategy for the determination of optimal point distribution is proposed. In the suggested adaptive refinement scheme, the local refinement indicators used are computed by considering the partition of unity property of the RKPM shape functions. In addition, a simple but effective variable support size definition scheme is suggested to ensure the robustness of the adaptive RKPM procedure. The performance of the suggested adaptive procedure is tested by using it to solve several benchmark problems. Numerical results indicated that the suggested refinement scheme can lead to the generation of nearly optimal meshes for both smooth and singular problems. The optimal convergence rate of the RKPM is restored and thus the effectivity indices of the Z–Z error estimator are converging to the ideal value of unity as the meshes are refined.     

Convergence,adaptive refinement,and scaling in 1D peridynamics

We introduce here adaptive refinement algorithms for the non‐local method peridynamics, which was proposed in (J. Mech. Phys. Solids 2000; 48 :175–209) as a reformulation of classical elasticity for discontinuities and long‐range forces. We use scaling of the micromodulus and horizon and discuss the particular features of adaptivity in peridynamics for which multiscale modeling and grid refinement are closely connected. We discuss three types of numerical convergence for peridynamics and obtain uniform convergence to the classical solutions of static and dynamic elasticity problems in 1D in the limit of the horizon going to zero. Continuous micromoduli lead to optimal rates of convergence independent of the grid used, while discontinuous micromoduli produce optimal rates of convergence only for uniform grids. Examples for static and dynamic elasticity problems in 1D are shown. The relative error for the static and dynamic solutions obtained using adaptive refinement are significantly lower than those obtained using uniform refinement, for the same number of nodes. Copyright © 2008 John Wiley & Sons, Ltd.     

An automatic adaptive refinement finite element procedure for 2D elastostatic analysis

An automatic adaptive refinement procedure for finite element analysis is presented. The procedure is applied to two-dimensional elastostatic problems to obtain solutions of prescribed accuracy. Through the combined use of new mesh generator using contour developed by Lo¹ and the concept of strain energy concentration, high-quality graded finite element meshes are generated. The whole process is fully automatic and no user intervention is required during the successive cycles of the mesh refinements. The Zienkiewicz and Zhu² error estimator is found to be effective and has been adopted for the present implementation. In the numerical examples tested, the error estimator gives an accurate error norm estimation and the effectivity index of the estimator converges to a value close to unity.     

An automatic adaptive refinement procedure using triangular and quadrilateral meshes

An automatic adaptive refinement procedure for finite element analysis for two-dimensional stress analysis problems is presented. Through the combined use of the new mesh generator developed by the authors (to appear) for adaptive mesh generation and the Zienkiewicz-Zhu [Int. J. numer. Meth. Engng31, 1331–1382 (1992)] error estimator based on the superconvergent patch recovery technique, an adaptive refinement procedure can be formulated which can achieve the aimed accuracy very economically in one or two refinement steps. A simple method is also proposed to locate the existence and the position of singularities in the problem domain. Hence, little or no a priori knowledge about the location and strength of the singularities is required. The entire adaptive refinement procedure has been made fully automatic and no user intervention during successive cycles of mesh refinements is needed. The robustness and reliability of the refinement procedure have been tested by solving difficult practical problems involving complex domain geometry with many singularities. We found that in all the examples studied, regardless of the types of meshes employed, triangular and quadrilateral meshes, nearly optimal overall convergence rate is always achieved.     

Three‐dimensional grayscale‐free topology optimization using a level‐set based r‐refinement method

In this paper, we propose a three‐dimensional (3D) grayscale‐free topology optimization method using a conforming mesh to the structural boundary, which is represented by the level‐set method. The conforming mesh is generated in an r‐refinement manner; that is, it is generated by moving the nodes of the Eulerian mesh that maintains the level‐set function. Although the r‐refinement approach for the conforming mesh generation has many benefits from an implementation aspect, it has been considered as a difficult task to stably generate 3D conforming meshes in the r‐refinement manner. To resolve this task, we propose a new level‐set based r‐refinement method. Its main novelty is a procedure for minimizing the number of the collapsed elements whose nodes are moved to the structural boundary in the conforming mesh; in addition, we propose a new procedure for improving the quality of the conforming mesh, which is inspired by Laplacian smoothing. Because of these novelties, the proposed r‐refinement method can generate 3D conforming meshes at a satisfactory level, and 3D grayscale‐free topology optimization is realized. The usefulness of the proposed 3D grayscale‐free topology optimization method is confirmed through several numerical examples. Copyright © 2017 John Wiley & Sons, Ltd.     

Automatic adaptive FE analysis of thin‐walled structures using 3D solid elements

In this study, a new automatic adaptive refinement procedure for thin‐walled structures using 3D solid elements is suggested. This procedure employs a specially designed superconvergent patch recovery (SPR) procedure for stress recovery, the Zienkiewicz and Zhu (Z–Z) error estimator for the a posteriori error estimation, a new refinement strategy for new element size prediction and a special mesh generator for adaptive mesh generation. The proposed procedure is different from other schemes in such a way that the problem domain is separated into two distinct parts: the shell part and the junction part. For stress recovery and error estimation in the shell part, special nodal coordinate systems are used and the stress field is separated into two components. For the refinement strategy, different procedures are employed for the estimation of new element sizes in the shell and the junction parts. Numerical examples are given to validate the effectiveness of the suggested procedure. It is found that by using the suggested refinement procedure, when comparing with uniform refinement, higher convergence rates were achieved and more accurate final solutions were obtained by using fewer degrees of freedoms and less amount of computational time. Copyright © 2008 John Wiley & Sons, Ltd.     

Hierarchical mesh adaptation of 2D quadrilateral elements

In this paper, we present some examples which illustrate the use of adaptive finite element analysis using a new hierarchical refinement algorithm for 2D quadrilateral elements. Projection type a posteriori error estimators based on a global stress smoothing and a simple stress averaging scheme have been used. The algorithm produces well refined meshes for some simple examples which contain stress singularities and stress concentrations. The convergence of the energy norm error has been found to be improved over a previous simple splitting algorithm.     

A multi‐level adaptive mesh refinement method for level set simulations of multiphase flow on unstructured meshes

An adaptive mesh refinement (AMR) technique is proposed for level set simulations of incompressible multiphase flows. The present AMR technique is implemented for two‐dimensional/three‐dimensional unstructured meshes and extended to multi‐level refinement. Smooth variation of the element size is guaranteed near the interface region with the use of multi‐level refinement. A Courant–Friedrich–Lewy condition for zone adaption frequency is newly introduced to obtain a mass‐conservative solution of incompressible multiphase flows. Finite elements around the interface are dynamically refined using the classical element subdivision method. Accordingly, finite element method is employed to solve the problems governed by the incompressible Navier–Stokes equations, using the level set method for dynamically updated meshes. The accuracy of the adaptive solutions is found to be comparable with that of non‐adaptive solutions only if a similar mesh resolution near the interface is provided. Because of the substantial reduction in the total number of nodes, the adaptive simulations with two‐level refinement used to solve the incompressible Navier–Stokes equations with a free surface are about four times faster than the non‐adaptive ones. Further, the overhead of the present AMR procedure is found to be very small, as compared with the total CPU time for an adaptive simulation. Copyright © 2016 John Wiley & Sons, Ltd.     

New approaches for error estimation and adaptivity for 2D potential boundary element methods

This work presents two new error estimation approaches for the BEM applied to 2D potential problems. The first approach involves a local error estimator based on a gradient recovery procedure in which the error function is generated from differences between smoothed and non‐smoothed rates of change of boundary variables in the local tangential direction. The second approach involves the external problem formulation and gives both local and global measures of error, depending on a choice of the external evaluation point. These approaches are post‐processing procedures. Both estimators show consistency with mesh refinement and give similar qualitative results. The error estimator using the gradient recovery approach is more general, as this formulation does not rely on an ‘optimal’ choice of an external parameter. This work presents also the use of a local error estimator in an adaptive mesh refinement procedure. This r‐refinement approach is based on the minimization of the standard deviation of the local error estimate. A non‐linear programming procedure using a feasible‐point method is employed using Lagrange multipliers and a set of active constraints. The optimization procedure produces finer meshes close to a singularity and results that are consistent with the problem physics. Copyright © 2002 John Wiley & Sons, Ltd.     

Meshfree natural vibration analysis of 2D structures

Determination of resonance frequencies and vibration modes of mechanical structures is one of the most important tasks in the product design procedure. The main goal of this paper is to describe a pioneering application of the solution structure method (SSM) to 2D structural natural vibration analysis problems and investigate the numerical properties of the method. SSM is a meshfree method which enables construction of the solutions to the engineering problems that satisfy exactly all prescribed boundary conditions. This method is capable of using spatial meshes that do not conform to the shape of a geometric model. Instead of using the grid nodes to enforce boundary conditions, it employs distance fields to the geometric boundaries and combines them with the basis functions and prescribed boundary conditions at run time. This defines unprecedented geometric flexibility of the SSM as well as the complete automation of the solution procedure. In the paper we will explain the key points of the SSM as well as investigate the accuracy and convergence of the proposed approach by comparing our results with the ones obtained using analytical methods or traditional finite element analysis. Despite in this paper we are dealing with 2D in-plane vibrations, the proposed approach has a straightforward generalization to model vibrations of 3D structures.     

A combined r-h adaptive strategy based on material forces and error assessment for plane problems and bimaterial interfaces

An r-h adaptive scheme has been proposed and formulated for analysis of bimaterial interface problems using adaptive finite element method. It involves a combination of the configurational force based r-adaption with weighted laplacian smoothing and mesh enrichment by h-refinement. The Configurational driving force is evaluated by considering the weak form of the material force balance for bimaterial inerface problems. These forces assembled at nodes act as an indicator for r-adaption. A weighted laplacian smoothing is performed for smoothing the mesh. The h-adaptive strategy is based on a modifed weighted energy norm of error evaluated using supercovergent estimators. The proposed method applies specific non sliding interface strain compatibility requirements across inter material boundaries consistent with physical principles to obtain modified error estimators. The best sequence of combining r- and h-adaption has been evolved from numerical study. The study confirms that the proposed combined r-h adaption is more efficient than a purely h-adaptive approach and more flexible than a purely r-adaptive approach with better convergence characteristics and helps in obtaining optimal finite element meshes for a specified accuracy.     

基于EEP超收敛解的自适应有限元法特性分析

基于EEP （单元能量投影）超收敛计算的自适应有限元法,已对一系列问题取得成功,但其自适应特性尚缺乏相关研究。该文以二阶常微分方程为模型问题,同时考察基于EEP和SPR （超收敛分片恢复）超收敛解的自适应分析方法,与有限元最优网格进行了比较分析,进而提出反映自适应有限元收敛特性的估计式,并给出了自适应收敛率β的定义。该文给出的数值试验表明:采用m次单元,对于解答光滑的问题,SPR法与EEP法均可有效用于自适应求解,其位移可按最大模获得m+1的自适应收敛率;对于奇异因子为α（<1）的奇异问题,SPR法失效,而基于EEP法的自适应求解,其位移按最大模可获得m+α的自适应收敛率,远高于α的常规有限元收敛率。     

A new automatic adaptive 3D solid mesh generation scheme for thin‐walled structures

A new algorithm to generate three‐dimensional (3D) mesh for thin‐walled structures is proposed. In the proposed algorithm, the mesh generation procedure is divided into two distinct phases. In the first phase, a surface mesh generator is employed to generate a surface mesh for the mid‐surface of the thin‐walled structure. The surface mesh generator used will control the element size properties of the final mesh along the surface direction. In the second phase, specially designed algorithms are used to convert the surface mesh to a 3D solid mesh by extrusion in the surface normal direction of the surface. The extrusion procedure will control the refinement levels of the final mesh along the surface normal direction. If the input surface mesh is a pure quadrilateral mesh and refinement level in the surface normal direction is uniform along the whole surface, all hex‐meshes will be produced. Otherwise, the final 3D meshes generated will eventually consist of four types of solid elements, namely, tetrahedron, prism, pyramid and hexahedron. The presented algorithm is highly flexible in the sense that, in the first phase, any existing surface mesh generator can be employed while in the second phase, the extrusion procedure can accept either a triangular or a quadrilateral or even a mixed mesh as input and there is virtually no constraint on the grading of the input mesh. In addition, the extrusion procedure development is able to handle structural joints formed by the intersections of different surfaces. Numerical experiments indicate that the present algorithm is applicable to most practical situations and well‐shaped elements are generated. Copyright © 2004 John Wiley & Sons, Ltd.     

On the development of adaptive random differential quadrature method with an error recovery technique and its application in the locally high gradient problems

An adaptive numerical method called, the adaptive random differential quadrature (ARDQ) method is presented in this paper. In the ARDQ method, the random differential quadrature (RDQ) method is coupled with a posteriori error estimator based on relative error norm in the displacement field. An error recovery technique, based on the least square averaging over the local interpolation domain, is proposed which improves the solution accuracy as the spacing, h → 0. In the adaptive refinement, a novel convex hull approach with the vectors cross product is proposed to ensure that the newly created nodes are always within the computational domain. The ARDQ method numerical accuracy is successfully evaluated by solving several 1D, 2D and irregular domain problems having locally high gradients. It is concluded from the convergence values that the ARDQ method coupled with error recovery technique can be effectively used to solve the locally high gradient initial and boundary value problems.     

A posteriori error estimates and adaptive procedures for the meshless Galerkin boundary node method for 3D potential problems

The Galerkin boundary node method (GBNM) is a boundary only meshless method that combines variational formulations of boundary integral equations with the moving least-squares approximations. This paper presents the mathematical derivation of a posteriori error estimates and adaptive refinement procedures for the GBNM for 3D potential problems. Two types of error estimators are developed in detail. One is a perturbation error estimator that is formulated based on the difference between numerical solutions obtained using two successive nodal arrangements. The other is a projection error estimator that is formulated based on the difference between the GBNM solution itself and its L²-orthogonal projection. The reliability and efficiency of both types of error estimators is established. That is, these error estimators are proven to have an upper and a lower bound by the constant multiples of the exact error in the energy norm. A localization technique is introduced to accommodate the non-local property of integral operators for the needed local and computable a posteriori error indicators. Convergence analysis results of corresponding adaptive meshless procedures are also given. Numerical examples with high singularities illustrate the theoretical results and show that the proposed adaptive procedures are simple, effective and efficient.     

Voxel‐based meshing and unit‐cell analysis of textile composites

Unit‐cell homogenization techniques are frequently used together with the finite element method to compute effective mechanical properties for a wide range of different composites and heterogeneous materials systems. For systems with very complicated material arrangements, mesh generation can be a considerable obstacle to usage of these techniques. In this work, pixel‐based (2D) and voxel‐based (3D) meshing concepts borrowed from image processing are thus developed and employed to construct the finite element models used in computing the micro‐scale stress and strain fields in the composite. The potential advantage of these techniques is that generation of unit‐cell models can be automated, thus requiring far less human time than traditional finite element models. Essential ideas and algorithms for implementation of proposed techniques are presented. In addition, a new error estimator based on sensitivity of virtual strain energy to mesh refinement is presented and applied. The computational costs and rate of convergence for the proposed methods are presented for three different mesh‐refinement algorithms: uniform refinement; selective refinement based on material boundary resolution; and adaptive refinement based on error estimation. Copyright © 2003 John Wiley & Sons, Ltd.     

Automatic adaptive refinement for plate bending problems using Reissner–Mindlin plate bending elements

The influence of the presence of singular points and boundary layers associated with the edge effects in a Reissner–Mindlin (RM) plate in the design of an optimal mesh for a finite element solution is studied, and methods for controlling the discretization error of the solution are suggested. An effective adaptive refinement strategy for the solution of plate bending problems based on the RM plate bending model is developed. This two-stage adaptive strategy is designed to control both the total and the shear error norms of a plate in which both singular points and boundary layers are present. A series of three different order assumed strain RM plate bending elements has been used in the adaptive refinement procedure. The locations of optimal sampling points and the effect of element shape distortions on the theoretical convergence rate of these elements are given and discussed. Numerical experiments show that the suggested refinement procedure is effective and that optimally refined meshes can be generated. It is also found that all the plate bending elements used can attain their full convergence rates regardless of the presence of singular points and boundary layers inside the problem domain. Boundary layer effects are well captured in all the examples tested and the use of a second stage of refinement to control the shear error is justified. In addition, tests on the Zienkiewicz–Zhu error estimator show that their performances are satisfactory. Finally, tests of the relative effectiveness of the plate bending elements used have also been made and it is found that while the higher order cubic element is the most accurate element tested, the quadratic element tested is the most efficient one in terms of CPU time used. © 1998 John Wiley & Sons, Ltd.     

An adaptive IgA-BEM with hierarchical B-splines based on quasi-interpolation quadrature schemes

The isogeometric formulation of the boundary element method (IgA-BEM) is investigated within the adaptivity framework. Suitable weighted quadrature rules to evaluate integrals appearing in the Galerkin BEM formulation of 2D Laplace model problems are introduced. The proposed quadrature schemes are based on a spline quasi-interpolation (QI) operator and properly framed in the hierarchical setting. The local nature of the QI perfectly fits with hierarchical spline constructions and leads to an efficient and accurate numerical scheme. An automatic adaptive refinement strategy is driven by a residual-based error estimator. Numerical examples show that the optimal convergence rate of the Galerkin solution is recovered by the proposed adaptive method.     

Extended isogeometric analysis based on PHT‐splines for crack propagation near inclusions

Adaptive local refinement is one of the main issues for isogeometric analysis (IGA). In this paper, an adaptive extended IGA (XIGA) approach based on polynomial splines over hierarchical T‐meshes (PHT‐splines) for modeling crack propagation is presented. The PHT‐splines overcome certain limitations of nonuniform rational B‐splines–based formulations; in particular, they make local refinements feasible. To drive the adaptive mesh refinement, we present a recovery‐based error estimator for the proposed method. The method is based on the XIGA method, in which discontinuous enrichment functions are added to the IGA approximation and this method does not require remeshing as the cracks grow. In addition, crack propagation is modeled by successive linear extensions that are determined by the stress intensity factors under linear elastic fracture mechanics. The proposed method has been used to analyze numerical examples, and the stress intensity factors results were compared with reference results. The findings demonstrate the accuracy and efficiency of the proposed method.