Superconvergent Patch Recovery Technique Research Articles

Topology optimization of fluid‐structure interaction problems with total stress equilibrium

AbstractThis work extends force coupling in the topology optimization of fluid‐structure interaction problems from hydrostatic to total stresses through the inclusion of viscous stress components. The superconvergent patch recovery technique is implemented to remove the discontinuities in velocity derivatives over the finite elements boundaries. The sensitivity analysis is derived analytically for the superconvergent patch recovery approach and further verified through the use of the complex‐step derivative approximation method. Numerical examples demonstrate a differentiation in the optimized designs using pressure versus total stress coupling depending on the flow characteristics of the design problem.

Application of machine learning in efficient stress recovery in finite element analysis

In this work, artificial neural network (ANN), a machine learning technique is applied in the field of stress recovery in finite element analysis (FEA). The most crucial design factor in structural analysis is stress. However, stresses calculated directly from FEA, are not accurate except at a few points called the super-convergent points. The stresses at the super-convergent points are targeted to map stress field using ANN for the whole domain. A higher-order element is used to calculate the reference stress values. The errors are calculated in the absolute sense and the results are compared with the superconvergent patch recovery (SPR) technique and also with the directly calculated stresses. Multiple mesh sizes are used to check the convergence with mesh refinement. The Cook’s skew beam problem is solved to apply the stress recovery using ANN.

Improvement of the Zienkiewicz–Zhu Error Recovery Technique Using a Patch Configuration

The Zienkiewicz–Zhu (ZZ) super-convergent patch recovery technique based on a node neighborhood patch configuration is used most widely for recovery of the stress field of a finite element analysis. In this study, an improved ZZ recovery technique using element neighborhood patch configuration is proposed. The improved recovery procedure is based on recovery of the stress field in the least-squares sense over an element patch that consists of the union of the elements surrounding the element under consideration. The proposed patch configuration provides more sampling points and improves the performance of the standard ZZ recovery technique. The effectiveness and reliability of the improved ZZ recovery approach is demonstrated through plane elastic and plastic plate problems. The problem domain is discretized with triangular and quadrilateral elements of different sizes. A comparison of the quality of error estimation using the ZZ recovery of derivative field and recovery of the displacement field using similar element neighborhood patch configurations is also presented. The numerical results show that the ZZ recovery technique and the displacement recovery technique, using a modified patch configuration, yield better results, convergence rate, and effectivity as compared with the standard ZZ super-convergent patch recovery technique. It is concluded that the improved ZZ recovery technique-based adaptive finite element analysis is very effective for converging a predefined accuracy with a significantly smaller number of degrees of freedom, especially in an elastic problem. It is also concluded that the improved ZZ recovery technique captures the plastic deformation problem solution errors more reliably than the standard ZZ recovery technique.

An efficient stress recovery technique in adaptive finite element method using artificial neural network

In this paper, an efficient stress recovery technique is presented to estimate the recovered stress field at the nodal points. The feed–forward back–propagation multilayer perceptron (MLP) neural network approach is employed to improve the accuracy of the stress recovery method. An automatic adaptive mesh refinement is performed based on a–posteriori Zienkiewicz–Zhu error estimation method. The proposed technique is employed to recover the stress field accurately in the regions with a high stress gradient where the conventional recovery techniques are not able to improve the stress fields efficiently due to the singular behavior of problem. Finally, several numerical examples are solved to demonstrate the efficiency and accuracy of the proposed computational algorithm. The results are compared with the conventional methods, including the averaging method, superconvergent patch recovery (SPR) technique, and weighted superconvergent patch recovery (WSPR) method that illustrates how the artificial neural network can be used accurately to recover the stress field.

Superconvergent patch recovery with constraints for three‐dimensional contact problems within the Cartesian grid Finite Element Method

SummaryThe superconvergent patch recovery technique with constraints (SPR‐C) consists in improving the accuracy of the recovered stresses obtained with the original SPR technique by considering known information about the exact solution, like the internal equilibrium equation, the compatibility equation or the Neumann boundary conditions, during the recovery process. In this paper the SPR‐C is extended to consider the equilibrium around the contact area when solving contact problems with the Cartesian grid Finite Element Method. In the proposed method, the Finite Element stress fields of both bodies in contact are considered during the recovery process and the equilibrium is enforced by means of the continuity of tractions along the contact surface.

Adaptive extended isogeometric analysis based on PHT-splines for thin cracked plates and shells with Kirchhoff–Love theory

• We implement an extended isogeometric analysis based on PHT-splines for thin-walled structures fracture problems. • C 1 -continuity is guaranteed by high smoothness of PHT-splines thus preventing hinge effects in Kirchhoff–Love theory. • We present a stress recovery-based posteriori error estimator technique to drive the adaptive procedure in cracked thin structures. • Results obtained using the adaptive XIGA-PHT demonstrate efficiency and accuracy compared with available literature solutions. In this paper, a posteriori error estimation and mesh adaptation approach for thin plate and shell structures of through-the-thickness crack is presented. This method uses the extended isogeometric analysis (XIGA) based on PHT-splines (Polynomial splines over Hierarchical T-meshes), which is abbreviated as XIGA-PHT. In XIGA-PHT, the isogeometric displacement approximation is locally enriched with enrichment functions, which efficiently capture the displacement discontinuity across the crack face as well as the stress singularity in the vicinity of the crack tip. On the one hand, the rotational degrees of freedom (RDOFs) are not required in Kirchhoff–Love theory, which drastically reduces the complexity of enrichment mode and computational scale for crack analysis. On the other hand, the PHT-splines basis functions can automatically satisfy the requirement of C 1 -continuity for the Kirchhoff–Love theory. Moreover, the PHT-splines facilitate the local refinement, which is the deficiency of NURBS-based isogeometric formulations. The local refinement is highly suitable for adaptive analysis. The stress recovery-based posteriori error estimator combined with the superconvergent patch recovery (SPR) technique is used to evaluate the approximate local discretization error. A new strategy for selecting enriched recovered functions in the enriched areas was proposed. Special functions extracted from the asymptotic stress solutions are applied to obtain the recovered stress field in the enriched area. The results of stress intensity factors or J-integral values obtained by the adaptive XIGA-PHT are compared with reference solutions. Several thin plate and shell illustrative examples demonstrate the effectiveness and accuracy of the proposed adaptive XIGA-PHT.

Improvements on a non-invasive, parameter-free approach to inverse form finding

Our objective is to determine the optimal undeformed workpiece geometry (material configuration) within forming processes when the prescribed deformed geometry (spatial configuration) is given. For solving the resulting shape optimization problem—also denoted as inverse form finding—we use a novel parameter-free approach, which relocates in each iteration the material nodal positions as design variables. The spatial nodal positions computed by an elasto-plastic finite element (FE) forming simulation are compared with their prescribed values. The objective function expresses a least-squares summation of the differences between the computed and the prescribed nodal positions. Here, a recently developed shape optimization approach (Landkammer and Steinmann in Comput Mech 57(2):169–191, 2016) is investigated with a view to enhance its stability and efficiency. Motivated by nonlinear optimization theory a detailed justification of the algorithm is given. Furthermore, a classification according to shape changing design, fixed and controlled nodal coordinates is introduced. Two examples with large elasto-plastic strains demonstrate that using a superconvergent patch recovery technique instead of a least-squares ( $$L^{2}$$ )-smoothing improves the efficiency. Updating the interior discretization nodes by solving a fictitious elastic problem also reduces the number of required FE iterations and avoids severe mesh distortions. Furthermore, the impact of the inclusion of the second deformation gradient in the Hessian of the Quasi-Newton approach is analyzed. Inverse form finding is a crucial issue in metal forming applications. As a special feature, the approach is designed to be coupled in a non-invasive fashion to arbitrary FE software.

Numerical Solution of Two Phase Solidification Problem Using Dynamic Substructuring Based on Adaptive Error Estimation

Numerical solution of solidification of metals with a sharp front, in particular solidification of lead, is investigated. Considering the fact that the associated CPU time and memory requirement may be costly for large domains, alternatives are searched. It is observed that using a substructuring technique with a local mesh refinement is promising. Following, by the use of an adaptive error estimation algorithm to find the location of solidification front and mushy zone, dynamic substructuring technique is developed to decrease the computational cost and to increase the accuracy of results. Superconvergent patch recovery technique is used to obtain the heat fluxes to evaluate the error energy norm of elements at each analysis step. Solidification front, mushy zone and elements having errors above a threshold value are captured with the error estimator. Then, elements having errors above the threshold value are refined by creating a substructure which is independent from the original global mesh. Equations of the global coarse mesh are augmented with the equations of the substructure. Employing the equations of the original coarse mesh help reduce the computational cost. Numerical solutions are presented and it is shown that the proposed approach has advantages over the alternative methods and, by the virtue of the adaptive error estimation algorithm, significantly decreases the CPU time of numerical solutions while it increases the accuracy of solutions and locates precisely the solidification front and mushy zone. Keywords: Solidification; Finite Element Methods; Substructuring; Stefan Problem; Error Estimation DOI: 10.17350/HJSE19030000017 Full Text:

Mesh adaptivity driven by goal-oriented locally equilibrated superconvergent patch recovery

Goal-oriented error estimates (GOEE) have become popular tools to quantify and control the local error in quantities of interest (QoI), which are often more pertinent than local errors in energy for design purposes (e.g. the mean stress or mean displacement in a particular area, the stress intensity factor for fracture problems). These GOEE are one of the key unsolved problems of advanced engineering applications in, for example, the aerospace industry. This work presents a simple recovery-based error estimation technique for QoIs whose main characteristic is the use of an enhanced version of the Superconvergent Patch Recovery (SPR) technique previously used for error estimation in the energy norm. This enhanced SPR technique is used to recover both the primal and dual solutions. It provides a nearly statically admissible stress field that results in accurate estimations of the local contributions to the discretisation error in the QoI and, therefore, in an accurate estimation of this magnitude. This approach leads to a technique with a reasonable computational cost that could easily be implemented into already available finite element codes, or as an independent postprocessing tool.

Crack growth modeling via 3D automatic adaptive mesh refinement based on modified-SPR technique

In this paper, the three-dimensional automatic adaptive mesh refinement is presented in modeling the crack propagation based on the modified superconvergent patch recovery technique. The technique is developed for the mixed mode fracture analysis of different fracture specimens. The stress intensity factors are calculated at the crack tip region and the crack propagation is determined by applying a proper crack growth criterion. An automatic adaptive mesh refinement is employed on the basis of modified superconvergent patch recovery (MSPR) technique to simulate the crack growth by applying the asymptotic crack tip solution and using the collapsed quarter-point singular tetrahedral elements at the crack tip region. A-posteriori error estimator is used based on the Zienkiewicz–Zhu method to estimate the error of fracture parameters and predict the crack path pattern. Finally, the efficiency and accuracy of proposed computational algorithm is demonstrated by several numerical examples.

Numerical simulation of multiple crack growth in brittle materials with adaptive remeshing

AbstractIn this paper, an automated adaptive remeshing procedure is presented for simulation of arbitrary shape crack growth in multiple cracked bodies. The Zienkiewicz–Zhu error estimator is employed in conjunction with the modified superconvergent patch recovery (SPR) technique to obtain more accurate estimation of error. A stability analysis is performed to determine active cracks from a set of competitive cracks. Various crack growth criteria together with the respective crack trajectory prediction are compared. Several numerical examples are illustrated to demonstrate the efficiency, robustness and accuracy of computational algorithm in the simulation of multiple crack growth. Copyright © 2010 John Wiley & Sons, Ltd.

Three-dimensional data transfer operators in large plasticity deformations using modified-SPR technique

In this paper, the data transfer operators are developed in 3D large plasticity deformations using superconvergent patch recovery (SPR) method. The history-dependent nature of plasticity problems necessitates the transfer of all relevant variables from the old mesh to new one, which is performed in three main stages. In the first step, the history-dependent internal variables are transferred from the Gauss points of old mesh to nodal points. The variables are then transferred from nodal points of old mesh to nodal points of new mesh. Finally, the values are computed at the Gauss points of new mesh using their values at nodal points. As the solution procedure, in general, cannot be re-computed from the initial configuration, it is continued from the previously computed state. In particular, the transfer operators are defined for mapping of the state and internal variables between different meshes. Aspects of the transfer operators are presented by fitting the best polynomial function with the C 0, C 1 and C 2 continuity in 3D superconvergent patch recovery technique. Finally, the efficiency of the proposed model and computational algorithms is demonstrated through numerical examples.

A recovery‐type error estimator for the extended finite element method based on singular+smooth stress field splitting

AbstractA new stress recovery procedure that provides accurate estimations of the discretization error for linear elastic fracture mechanic problems analyzed with the extended finite element method (XFEM) is presented. The procedure is an adaptation of the superconvergent patch recovery (SPR) technique for the XFEM framework. It is based on three fundamental aspects: (a) the use of a singular+smooth stress field decomposition technique involving the use of different recovery methods for each field: standard SPR for the smooth field and reconstruction of the recovered singular field using the stress intensity factor K for the singular field; (b) direct calculation of smoothed stresses at integration points using conjoint polynomial enhancement; and (c) assembly of patches with elements intersected by the crack using different stress interpolation polynomials at each side of the crack. The method was validated by testing it on problems with an exact solution in mode I, mode II, and mixed mode and on a problem without analytical solution. The results obtained showed the accuracy of the proposed error estimator. Copyright © 2008 John Wiley & Sons, Ltd.

Nodal velocity derivatives of finite element solutions: The FiND method for incompressible Navier–Stokes equations

AbstractThis paper presents the development and application of the finite node displacement (FiND) method to the incompressible Navier–Stokes equations. The method computes high‐accuracy nodal derivatives of the finite element solutions. The approach imposes a small displacement to individual mesh nodes and solves a very small problem on the patch of elements surrounding the node. The only unknown is the value of the solution (u, p) at the displaced node. A finite difference between the original and the perturbed values provides the directional derivative. Verification by grid refinement studies is shown for two‐dimensional problems possessing a closed‐form solution: a Poiseuille flow and a flow mimicking a boundary layer. For internal nodes, the method yields accuracy slightly superior to that of the superconvergent patch recovery (SPR) technique of Zienkiewicz and Zhu (ZZ). We also present a variant of the method to treat boundary nodes. The local discretization is enriched by inserting an additional mesh point very close to the boundary node of interest. Computations show that the resulting nodal derivatives are much more accurate than those obtained by the ZZ SPR technique. Copyright © 2008 John Wiley & Sons, Ltd.

Modeling of crack propagation via an automatic adaptive mesh refinement based on modified superconvergent patch recovery technique

In this paper, an automated adaptive remeshing procedure is presented for simulation of arbitrary shape crack growth in a 2D finite element mesh. The Zienkiewicz–Zhu error estimator is employed in conjunction with a modified SPR technique based on the recovery of gradients using analytical crack-tip fields in order to obtain more accurate estimation of errors. The optimization of crack-tip singular finite element size is achieved through the adaptive mesh strategy. Finally, several numerical examples are illustrated to demonstrate the effectiveness, robustness and accuracy of computational algorithm in calculation of fracture parameters and prediction of crack path pattern.

Three-dimensional superconvergent patch recovery method and its application to data transferring in small-strain plasticity

In this paper, a 3D Superconvergent Patch Recovery (SPR) method is developed for data transferring in elasto-plasticity. The transfer operators are presented for mapping of the state and internal variables between different meshes. In order to transfer the history-dependent variables from old mesh to new one, the internal variables are firstly mapped from the Gauss points to nodal points of old mesh, the variables are then transferred from nodal points of old mesh to nodal points of new mesh, and the values are finally transferred from the nodal points to Gauss points of new mesh. As the solution procedure cannot be re-computed from the initial state, it is continued from the previously computed state. Aspects of the transfer operators are presented in 3D superconvergent patch recovery technique by fitting the best polynomial function with C0, C1 and C2 continuity. Finally, the efficiency of the proposed model and computational algorithms is demonstrated by several numerical examples.

Derivative recovery and a posteriori error estimate for extended finite elements

This paper is the first attempt at error estimation for extended finite elements. The goal of this work is to devise a simple and effective local a posteriori error estimate for partition of unity enriched finite element methods such as the extended finite element method (XFEM). In each element, the local estimator is the L 2 norm of the difference between the raw XFEM strain field and an enhanced strain field computed by extended moving least squares (XMLS) derivative recovery obtained from the raw nodal XFEM displacements. The XMLS construction is tailored to the nature of the solution. The technique is applied to linear elastic fracture mechanics, in which near-tip asymptotic functions are added to the MLS basis. The XMLS shape functions are constructed from weight functions following the diffraction criterion to represent the discontinuity. The result is a very smooth enhanced strain solution including the singularity at the crack tip. Results are shown for two- and three-dimensional linear elastic fracture mechanics problems in mode I and mixed mode. The effectivity index of the estimator is close to 1 and improves upon mesh refinement for the studied near-tip problem. It is also shown that for the linear elastic fracture mechanics problems treated, the proposed estimator outperforms one of the superconvergent patch recovery technique of Zienkiewicz and Zhu, which is only C 0. Parametric studies of the general performance of the estimator are also carried out.

The superconvergence patch recovery technique and data transfer operators in 3D plasticity problems

A decade has passed since the first appearance of the superconvergent patch recovery (SPR) method introduced by Zienkiewicz and Zhu [The superconvergence patch recovery and a posteriori error estimates, part I: the recovery techniques, Int. J. Numer. Methods Eng. 33 (1992) 1331–1364; The superconvergence patch recovery and a posteriori error estimates, part II: error estimates and adaptivity, Int. J. Numer. Methods Eng. 33 (1992) 1365–1380; Superconvergence and the superconvergent patch recovery, Finite Elem. Anal. Des. 19 (1995) 11–23]. The method is now widely used in engineering practices for its robustness and efficiency in computer implementation. This paper presents an extension of the SPR technique to 3D nonlinear problems with its application to 3D data transferring of continuum nonlinear mechanics. The transfer operators are presented for mapping of the state and internal variables between different meshes. Aspects of the transfer operators are presented in 3D superconvergent path recovery technique based on C 0 , C 1 and C 2 continuity using tetrahedral elements. Finally, the efficiency of the proposed model and computational algorithms is demonstrated by several numerical examples.

Computation of accurate nodal derivatives of finite element solutions: the finite node displacement method

AbstractThis paper presents a new method for extracting high‐accuracy nodal derivatives from finite element solutions. The approach involves imposing a finite displacement to individual mesh nodes, and solving a very small problem on the patch of surrounding elements, whose only unknown is the value of the solution at the displaced node. A finite difference between the original and perturbed values provides the directional derivative. Verification is shown for a one‐dimensional diffusion problem with exact nodal solution and for two‐dimensional scalar advective–diffusive problems. For internal nodes the method yields accuracy slightly superior to that of the superconvergent patch recovery (SPR) technique of Zienkiewicz and Zhu (ZZ). We also present a variant of the method to treat boundary nodes. In this case, the local discretization is enriched by inserting an additional mesh point very close to the boundary node of interest. We show that the new method gives normal derivatives at boundary points that are consistent with the so‐called ‘auxiliary fluxes’. The resulting nodal derivatives are much more accurate than those obtained by the ZZ SPR technique. Copyright © 2007 Crown in the right of Canada. Published by John Wiley & Sons, Ltd.

Improvement of the superconvergent patch recovery technique by the use of constraint equations: the SPR‐C technique

AbstractThe superconvergent patch recovery (SPR) technique is widely used in the evaluation of a recovered stress field σ* from the finite element solution σfe. Several modifications of the original SPR technique have been proposed. A new improvement of the SPR technique, called SPR‐C technique (Constrained SPR), is presented in this paper. This new technique proposes the use of the appropriate constraint equations in order to obtain stress interpolation polynomials in the patch σ that locally satisfy the equations that should be satisfied by the exact solution. As a result the evaluated expressions for σ will satisfy the internal equilibrium and compatibility equations in the whole patch and the boundary equilibrium equation at least in vertex boundary nodes and, under certain circumstances, along the whole boundary of the patch coinciding with the boundary of the domain. The results show that the use of this technique considerably improves the accuracy of the recovered stress field σ* and therefore the local effectivity of the ZZ error estimator. Copyright © 2006 John Wiley & Sons, Ltd.