Towards automatic adaptive geometrically non-linear shell analysis. Part I: Implementation of an h-adaptive mesh refinement procedure

Effective methods leading to automated adaptive numerical solutions to geometrically non-linear shell-type problems are studied in this work. In particular, procedures for improving the accuracy, the reliability and the computational efficiency of the finite element solutions are of primary interest here. This is addressed using h-adaptive mesh refinement based on a posteriori error estimation, self-adaptive methods in global incremental/iterative processes, as well as smart algorithms and heuristic approaches based on methods of knowledge engineering. Seemless integration of h-adaptive finite element methods with adaptive step-length control makes it possible to maintain a prescribed accuracy while maintaining the solution efficiency without user intervention throughout the process of a non-linear analysis. Several examples illustrate the merit and potential of the approach studied herein and confirm the feasibility of developing an automatic adaptive environment for geometrically non-linear analysis of shell structures.     

Topology optimization using a continuous density field and adaptive mesh refinement

A new method of topology optimization is introduced in which a continuous material field is combined with adaptive mesh refinement. Using a continuous material field with different analysis and design meshes allows the method to produce optimal designs that are free of numerical artifacts like checkerboard patterns and material islands. Adaptive mesh refinement is then applied to both meshes to precisely locate the optimal boundary of the final structure. A Helmholtz‐type density filter is used to prevent the appearance of small topological features as the mesh refinement proceeds. Results are presented for several test problems, including problems with geometrically complex domain boundaries.     

Refined h-adaptive finite element procedure for large deformation geotechnical problems

Adaptive finite element procedures automatically refine, coarsen, or relocate elements in a finite element mesh to obtain a solution with a specified accuracy. Although a significant amount of research has been devoted to adaptive finite element analysis, this method has not been widely applied to nonlinear geotechnical problems due to their complexity. In this paper, the h-adaptive finite element technique is employed to solve some complex geotechnical problems involving material nonlinearity and large deformations. The key components of h-adaptivity including robust mesh generation algorithms, error estimators and remapping procedures are discussed. This paper includes a brief literature review as well as formulation and implementation details of the h-adaptive technique. Finally, the method is used to solve some classical geotechnical problems and results are provided to illustrate the performance of the method.     

Generalized finite element method in structural nonlinear analysis – a <Emphasis Type="Italic">p</Emphasis>-adaptive strategy

This paper is concerned with an extension of the generalized finite element method, GFEM, to nonlinear analysis and to the proposition of a p-adaptive strategy. The p-adaptivity is considered due to the nodal enrichment scheme of the method. Here, such scheme consists of multiplying the partition of unity functions by a set of polynomials. In a first part, the performance of the method in nonlinear analysis of a reinforced concrete beam with progressive damage is presented. The adaptive strategy is then proposed on basis of a control over the approximation error. Aiming to estimate the approximation error, the equilibrated element residual method is adapted to the GFEM and to the nonlinear approach. Then, global and local error measures are defined. A numerical example is presented outlining the effectivity index of the error estimator proposed. Finally, a p-adaptive procedure is described and its good performance is illustrated by a numerical example.The authors gratefully acknowledge the Conselho Nacional de Desenvolvimento Cientìifico e Tecnológico (CNPq) at Brazil.     

A p-adaptive scaled boundary finite element method based on maximization of the error decrease rate

This study enhances the classical energy norm based adaptive procedure by introducing new refinement criteria, based on the projection-based interpolation technique and the steepest descent method, to drive mesh refinement for the scaled boundary finite element method. The technique is applied to p-adaptivity in this paper, but extension to h- and hp-adaptivity is straightforward. The reference solution, which is the solution of the fine mesh formed by uniformly refining the current mesh, is used to represent the unknown exact solution. In the new adaptive approach, a projection-based interpolation technique is developed for the 2D scaled boundary finite element method. New refinement criteria are proposed. The optimum mesh is assumed to be obtained by maximizing the decrease rate of the projection-based interpolation error appearing in the current solution. This refinement strategy can be interpreted as applying the minimisation steepest descent method. Numerical studies show the new approach out-performs the conventional approach.     

自适应网格方法在Stokes问题形状最优控制中的应用

为了求解流体力学中的形状最优控制问题,本文提出了一种与最优化准则方法相耦合的自适应网格方法.优化的目标是使得流体流动的能量耗散达到最小,状态方程是Stokes问题.本算法可以在减少计算量的情况下,保证流体的界面达到较高的分辨率.最优化算法采用的是非常稳定的经典最优化准则方法,自适应网格的指示函数是通过材料分布的信息得到的.虽然本文只是考虑了Stokes问题,但所得算法可以用来解决很广泛的一类流体动力学中的形状或拓扑最优化问题.     

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.     

A posteriori error analysis and adaptive processes in the finite element method: Part I—error analysis

This is a paper presented in two parts dealing respectively with error analysis and adaptive processes applied to finite element calculations. Part I contains the basic theory and methods of deriving error estimates for second-order problems. Part II of the paper deals with the strategy for adaptive refinement and concentrates on the p-convergent methods. It is shown that an extremely high rate of convergence is reached in practical problems using such procedures. Applications to realistic stress analysis and potential problems are presented.     

An adaptive interface‐enriched generalized FEM for the treatment of problems with curved interfaces

An adaptive refinement scheme is presented to reduce the geometry discretization error and provide higher‐order enrichment functions for the interface‐enriched generalized FEM. The proposed method relies on the h‐adaptive and p‐adaptive refinement techniques to reduce the discrepancy between the exact and discretized geometries of curved material interfaces. A thorough discussion is provided on identifying the appropriate level of the refinement for curved interfaces based on the size of the elements of the background mesh. Varied techniques are then studied for selecting the quasi‐optimal location of interface nodes to obtain a more accurate approximation of the interface geometry. We also discuss different approaches for creating the integration sub‐elements and evaluating the corresponding enrichment functions together with their impact on the performance and computational cost of higher‐order enrichments. Several examples are presented to demonstrate the application of the adaptive interface‐enriched generalized FEM for modeling thermo‐mechanical problems with intricate geometries. The accuracy and convergence rate of the method are also studied in these example problems. Copyright © 2015 John Wiley & Sons, Ltd.     

3D‐based hp‐adaptive first‐order shell finite element for modelling and analysis of complex structures—Part 2: Application to structural analysis

The paper presents a 3D‐based adaptive first‐order shell finite element to be applied to hierarchical modelling and adaptive analysis of complex structures. The main feature of the element is that it is equipped with 3D degrees of freedom, while its mechanical model corresponds to classical first‐order shell theory. Other useful features of the element are its modelling and adaptive capabilities. The element is assigned to hierarchical modelling and hpq‐adaptive analysis of shell parts of complex structures consisting of solid, thick‐ and thin‐shell parts, as well as of transition zones, where h, p and q denote the mesh density parameter and the longitudinal and transverse orders of approximation, respectively. The proposed hp‐adaptive first‐order shell element can be joined with 3D‐based hpq‐adaptive hierarchical shell elements or 3D hpp‐adaptive solid elements by means of the family of 3D‐based hpq/hp‐ or hpp/hp‐adaptive transition elements. The main objective of the first part of our research, presented in the first part of the paper, was to provide non‐standard information on the original parts of the element algorithm. Here we describe the second part of the research, devoted to the methodology and results of the application of the element to various plate and shell problems. The main objective of this part is to verify algorithms of the element and to show its usefulness in modelling and adaptive analysis of shell and plate parts of complex structures. In order to do that, there is a presentation of the results of a comparative analysis of model plate and shell problems using the classical and our elements, and equidistributed and integrated Legendre shape functions. For the plate problem a comparison of the results obtained from the adaptive and non‐adaptive analysis is also included. Additionally, some advantages of the application of our element are shown through a comparative analysis of p‐convergence of the thin plate problem and an adaptive analysis of the exemplary complex structure. Copyright © 2006 John Wiley & Sons, Ltd.     

A posteriori error analysis and adaptive processes in the finite element method: Part II—adaptive mesh refinement

This is a paper presented in two parts dealing respectively with error analysis and adaptive processes applied to finite element calculations. Part I contains the basic theory and methods of deriving error estimates for second-order problems. Part II of the paper deals with the strategy for adaptive refinement and concentrates again on the p-convergent methods. It is shown that an extremely high rate of convergence is reached in practical problems using such procedures. Applications to realistic stress analysis and potential problems are presented.     

Dual boundary‐element method: Simple error estimator and adaptivity

This paper is concerned with the effective numerical implementation of the adaptive dual boundary‐element method (DBEM), for two‐dimensional potential problems. Two boundary integral equations, which are the potential and the flux equations, are applied for collocation along regular and degenerate boundaries, leading always to a single‐region analysis. Taking advantage on the use of non‐conforming parametric boundary‐elements, the method introduces a simple error estimator, based on the discontinuity of the solution across the boundaries between adjacent elements and implements the p, h and mixed versions of the adaptive mesh refinement. Examples of several geometries, which include degenerate boundaries, are analyzed with this new formulation to solve regular and singular problems. The accuracy and efficiency of the implementation described herein make this a reliable formulation of the adaptive DBEM. Copyright © 2011 John Wiley & Sons, Ltd.     

A p‐hierarchical adaptive procedure for the scaled boundary finite element method

This paper describes a p‐hierarchical adaptive procedure based on minimizing the classical energy norm for the scaled boundary finite element method. The reference solution, which is the solution of the fine mesh formed by uniformly refining the current mesh element‐wise one order higher, is used to represent the unknown exact solution. The optimum mesh is assumed to be obtained when each element contributes equally to the global error. The refinement criteria and the energy norm‐based error estimator are described and formulated for the scaled boundary finite element method. The effectivity index is derived and used to examine quality of the proposed error estimator. An algorithm for implementing the proposed p‐hierarchical adaptive procedure is developed. Numerical studies are performed on various bounded domain and unbounded domain problems. The results reflect a number of key points. Higher‐order elements are shown to be highly efficient. The effectivity index indicates that the proposed error estimator based on the classical energy norm works effectively and that the reference solution employed is a high‐quality approximation of the exact solution. The proposed p‐hierarchical adaptive strategy works efficiently. Copyright © 2007 John Wiley & Sons, Ltd.     

Adaptive h-, p- and hp-versions for boundary integral element methods

Recently very promising results in a so-called hp-version of the finite element method have been obtained. The basic idea is a balanced combination of mesh refinement and increase of the polynomial degree of the shape functions. This idea is applied to a boundary collocation method in this paper. The new method is compared with adaptive h- and p-versions and it is shown in numerical examples that even in the presence of singularities in the exact solution exponential convergence is obtained.     

A scaled boundary finite element based explicit topology optimization approach for three-dimensional structures

This article proposes an efficient approach for solving three-dimensional (3D) topology optimization problem. In this approach, the number of design variables in optimization as well as the number of degrees of freedom in structural response analysis can be reduced significantly. This is accomplished through the use of scaled boundary finite element method (SBFEM) for structural analysis under the moving morphable component (MMC)-based topology optimization framework. In the proposed method, accurate response analysis in the boundary region dictates the accuracy of the entire analysis. In this regard, an adaptive refinement scheme is developed where the refined mesh is only used in the boundary region while relating coarse mesh is used away from the boundary. Numerical examples demonstrate that the computational efficiency of 3D topology optimization can be improved effectively by the proposed approach.     

Adaptive superposition of finite element meshes in non‐linear transient solid mechanics problems

An s‐adaptive finite element procedure is developed for the transient analysis of 2‐D solid mechanics problems with material non‐linearity due to progressive damage. The resulting adaptive method simultaneously estimates and controls both the spatial error and temporal error within user‐specified tolerances. The spatial error is quantified by the Zienkiewicz–Zhu error estimator and computed via superconvergent patch recovery, while the estimation of temporal error is based on the assumption of a linearly varying third‐order time derivatives of the displacement field in conjunction with direct numerical time integration. The distinguishing characteristic of the s‐adaptive procedure is the use of finite element mesh superposition (s‐refinement) to provide spatial adaptivity. Mesh superposition proves to be particularly advantageous in computationally demanding non‐linear transient problems since it is faster, simpler and more efficient than traditional h‐refinement schemes. Numerical examples are provided to demonstrate the performance characteristics of the s‐adaptive method for quasi‐static and transient problems with material non‐linearity. Copyright © 2007 John Wiley & Sons, Ltd.     

An h-p- multigrid method for finite element analysis

The finite element method generates solutions to partial differential equations by minimizing a strain energy based functional. Strain energy based techniques for adaptive mesh refinements are not always effective, however. The adaptive refinement technique proposed in this paper uses strain energy but also incorporates advantages from the h- and p- finite element methods, the multigrid method and a Delaunay based mesh generation method. The refinement technique converged rapidly and was numerically efficient when applied to determining stress concentrations around the circular hole of a thick plate under tension.     

Adaptive analysis of thin shells using facet elements

An adaptive mesh refinement (AMR) procedure is used in static thin shell analysis using triangular facet shell elements. The procedure described herein uses the h-version of adaptive refinement based on an error estimate determined by using the best guess values of bending moments and membrane forces obtained from a previous solution. It includes the use of a relaxation factor to achieve better convergence. Some examples are presented to illustrate this method. The results obtained are compared with those of uniform mesh refinement (UMR).     

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.