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Local maximum entropy shape functions based FE-EFGM coupling
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Technical Report from Engineering and Computing Sciences at Durham University
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... A comprehensive review of different FE-EFGM coupling procedures up to 2005 can be found in [20]. A new coupling procedure for FE-EFGM has recently been proposed by the authors in [21] for linear elastic and geometrically nonlinear problems. In this coupling procedure, max-ent shape functions are used in the EFG region of the problem domain. ...
... where b e n is the value of the elastic left Cauchy-Green strain matrix at the end of previous increment and is obtained by rearranging (19) in terms of b and using ε = ε e n , where ε e n is the elastic strain from the previously converged load step. (21) can be used in (19) to calculate the trial elastic strain ε e tr , which is input to the constitutive model, the output of which includes elastic strain ε e , stress σ and consistent or algorithmic tangent D alg . A Newton-Raphson incremental-iterative procedure is used, i.e. load is applied incrementally in steps and convergence is sought for each increment, using ...
An automatic adaptive coupling procedure is proposed for the finite element method (FEM) and the element-free Galerkin method (EFGM) for linear elasticity and for problems with both material and geometrical nonlinearities. In this new procedure, initially the whole of the problem domain is modelled using the FEM. During an analysis, those finite elements which violate a predefined error measure are automatically converted to an EFG zone. This EFG zone can be further refined by adding nodes, thus avoiding computationally expensive FE remeshing. Local maximum entropy shape functions are used in the EFG zone of the problem domain for two reasons: their weak Kronecker delta property at the boundaries allows straightforward imposition of essential boundary conditions and also provides a natural way to couple the EFG and FE regions compared to the use of moving least squares basis functions. The Zienkiewicz and Zhu error estimation procedure with the superconvergent patch method for strains and stresses recovery is used in the FE region of the problem domain, while the Chung and Belytschko error estimation procedure is used in the EFG region.
... A review of the FEM and MM coupling can be found in Ref. [39]. Direct coupling without transition or interface elements between a FEM region and a MM region using the max-ent shape functions was also proposed in Ref. [40][41][42] by exploiting their weak Kronecker delta property. However, the direct coupling between the displacement field satisfying the Kronecker delta property in the FEM region with the displacement field only satisfying the weak Kronecker delta property in the MM region a priori leads to a discrepancy of the displacement field at the FEM-MM interface. ...
The Finite Element Method (FEM) suffers from important drawbacks in problems involving excessive deformation of elements despite being universally applied to a wide range of engineering applications. While dynamic remeshing is often offered as the ideal solution, its computational cost, numerical noise and mathematical limitations in complex geometries are impeding its widespread use. Meshless methods (MM), however, by not relying on mesh connectivity, circumvent some of these limitations, while remaining computationally more expensive than the classic FEM. These problems in MM can be improved by coupling with FEM in a FEM-MM scheme, in which MM is used within sensitive regions that undergo large deformations while retaining the more efficient FEM for other less distorted regions. Here, we present a numerical framework combining the benefits of FEM and MM to study large deformation scenarios without heavily compromising on computational efficiency. In particular, the latter is maintained through two mechanisms: (1) coupling of FEM and MM discretisation schemes within one problem, which limits MM discretisation to domains that cannot be accurately modelled in FEM, and (2) a simplified MM parallelisation approach which allows for highly efficient speed-up. The proposed approach treats the problem as a quadrature point driven problem, thus making the treatment of the constitutive models, and thus the matrix and vector assembly fully method-agnostic. The MM scheme considers the maximum entropy (max-ent) approximation, in which its weak Kronecker delta property is leveraged in parallel calculations by convexifying the subdomains, and by refining meshes at the boundary in such a way that the higher density of nodes is mainly concentrated within the bulk of the domain. The latter ensures obtaining the Kronecker delta property at the boundary of the MM domain. The results, demonstrated by means of a few applications, show an excellent scalability and a good balance between accuracy and computational cost.
... The LME approximants have already found a large number of applications [22,23,24,25,26]. LME are smooth and nonnegative approximants with local support that possess the weak Kronecker-delta property in the sense that basis functions on the boundary are not influenced by internal nodes. ...
We present the cell‐based maximum entropy (CME) approximants in E³ space by constructing the smooth approximation distance function to polyhedral surfaces. CME is a meshfree approximation method combining the properties of the maximum entropy approximants and the compact support of element‐based interpolants. The method is evaluated in problems of large strain elastodynamics for three‐dimensional (3D) continua using the well‐established meshless total Lagrangian explicit dynamics method. The accuracy and efficiency of the method is assessed in several numerical examples in terms of computational time, accuracy in boundary conditions imposition, and strain energy density error. Due to the smoothness of CME basis functions, the numerical stability in explicit time integration is preserved for large time step. The challenging task of essential boundary condition (EBC) imposition in noninterpolating meshless methods (eg, moving least squares) is eliminated in CME due to the weak Kronecker‐delta property. The EBCs are imposed directly, similar to the finite element method. CME is proven a valuable alternative to other meshless and element‐based methods for large‐scale elastodynamics in 3D. A naive implementation of the CME approximants in E³ is available to download at https://www.mountris.org/software/mlab/cme.
... LME approximants use generalized barycentric coordinates based on Jayne's principle of maximum entropy [21] and provide a seamless transition between non-local approximation and simplicial interpolation on a Delaunay triangulation (linear interpolants in the context of FEM). LME approximants have already found a large number of applications [22,23,24,25,26]. LME approximants are smooth and nonnegative approximants with local support that possess the weak Kronecker-delta property in the sense that basis functions on the boundary are not influenced by internal nodes. ...
In this paper, we extend the Cell-based Maximum Entropy (CME) approximants in E3 by constructing smooth approximation distance function to polyhedral surfaces. The motivation of this work is to evaluate the CME approximants in the context of large strain elastodynamics for three-dimensional solids using the well-established Meshless Total Lagrangian Explicit Dynamics (MTLED) method. Several numerical examples are solved to evaluate the performance of CME in MTLED for both regular and irregular three-dimensional geometries in terms of computational time, accuracy in boundary conditions imposition, and errors in strain energy. The smoothness and the weak-Kronecker delta properties of CME basis functions result to long explicit time integration step and exact imposition of essential boundary conditions. These properties support the application of the proposed scheme in large-scale three-dimensional domains of arbitrary shape.
Purpose
– A variety of meshless methods have been developed in the last 20 years with an intention to solve practical engineering problems, but are limited to small academic problems due to associated high computational cost as compared to the standard finite element methods (FEM). The purpose of this paper is to develop an efficient and accurate algorithms based on meshless methods for the solution of problems involving both material and geometrical nonlinearities.
Design/methodology/approach
– A parallel two-dimensional linear elastic computer code is presented for a maximum entropy basis functions based meshless method. The two-dimensional algorithm is subsequently extended to three-dimensional adaptive nonlinear and three-dimensional parallel nonlinear adaptively coupled finite element, meshless method cases. The Prandtl-Reuss constitutive model is used to model elasto-plasticity and total Lagrangian formulations are used to model finite deformation. Furthermore, Zienkiewicz and Zhu and Chung and Belytschko error estimation procedure are used in the FE and meshless regions of the problem domain, respectively. The message passing interface library and open-source software packages, METIS and MUltifrontal Massively Parallel Solver are used for the high performance computation.
Findings
– Numerical examples are given to demonstrate the correct implementation and performance of the parallel algorithms. The agreement between the numerical and analytical results in the case of linear elastic example is excellent. For the nonlinear problems load-displacement curve are compared with the reference FEM and found in a very good agreement. As compared to the FEM, no volumetric locking was observed in the case of meshless method. Furthermore, it is shown that increasing the number of processors up to a given number improve the performance of parallel algorithms in term of simulation time, speedup and efficiency.
Originality/value
– Problems involving both material and geometrical nonlinearities are of practical importance in many engineering applications, e.g. geomechanics, metal forming and biomechanics. A family of parallel algorithms has been developed in this paper for these problems using adaptively coupled finite element, meshless method (based on maximum entropy basis functions) for distributed memory computer architectures.
We present a method for the automatic adaption of the support size of meshfree basis functions in the context of the numerical approximation of boundary value problems stemming from a minimum principle. The method is based on a variational approach, and the central idea is that the variational principle selects both the discretized physical fields and the discretization parameters, here those defining the support size of each basis function. We consider local maximum-entropy approximation schemes, which exhibit smooth basis functions with respect to both space and the discretization parameters (the node location and the locality parameters). We illustrate by the Poisson, linear and non-linear elasticity problems the effectivity of the method, which produces very accurate solutions with very coarse discretizations and finds unexpected patterns of the support size of the shape functions.
We present a method to process embedded smooth manifolds using sets of points alone. This method avoids any global parameterization and hence is applicable to surfaces of any genus. It combines three ingredients: (1) the automatic detection of the local geometric structure of the manifold by statistical learning methods; (2) the local parameterization of the surface using smooth meshfree (here maximum‐entropy) approximants; and (3) patching together the local representations by means of a partition of unity. Mesh‐based methods can deal with surfaces of complex topology, since they rely on the element‐level parameterizations, but cannot handle high‐dimensional manifolds, whereas previous meshfree methods for thin shells consider a global parametric domain, which seriously limits the kinds of surfaces that can be treated. We present the implementation of the method in the context of Kirchhoff–Love shells, but it is applicable to other calculations on manifolds in any dimension. With the smooth approximants, this fourth‐order partial differential equation is treated directly. We show the good performance of the method on the basis of the classical obstacle course. Additional calculations exemplify the flexibility of the proposed approach in treating surfaces of complex topology and geometry.
This thesis is concerned with the theoretical development and numerical implementation of efficient constitutive models for the analysis of particulate media (specifically clays) in structures undergoing geometrically non-linear behaviour. The Mohr-Coulomb and modified Cam-clay constitutive models have both been examined and extended to provide greater realism. Findings from this thesis will interest engineers working in numerical methods in solid mechanics, along with those investigating continuum mechanics, inelastic constitutive modelling and large strain plasticity. Although focused on soil plasticity, this research has relevance to other areas, such as metal forming and bio-engineering.
Initially the concepts of material and geometric non-linearity are reviewed. A general implicit backward Euler stress integration algorithm is detailed, including the derivation of the algorithmic consistent tangent. A framework for the analysis of anisotropic finite deformation elasto-plasticity is presented and a full incremental finite-element formulation provided. The first constitutive model developed in this thesis is a non-associated frictional perfect plasticity model based on a modified Reuleaux triangle. It is shown, through comparison with experimental data, that this model has advantages over the classical Mohr-Coulomb and Drucker-Prager models whilst still allowing for analytical implicit stress integration. An isotropic hyperplastic family of models which embraces the concept of a Critical State is then developed. This family is extended to include inelastic behaviour within the conventional yield surface and a Lode angle dependency on the anisotropic yield function which maintains convexity of both the surface and uniqueness of the Critical State cone. A calibration procedure is described and the integration and linearisation of the constitutive relations are detailed. All of the developed models are compared with established experimental data. Finally the models
are verified for use within finite deformation finite-element analyses. The importance of deriving the algorithmic consistent tangent is demonstrated and the influence of varying levels of model sophistication assessed in terms of both global behaviour and simulation run-time.
As we attempt to solve engineering problems of ever increasing complexity, so must we develop and learn new methods for doing so. The Finite Difference Method used for centuries eventually gave way to Finite Element Methods (FEM), which better met the demands for flexibility, effectiveness, and accuracy in problems involving complex geometry. Now, however, the limitations of FEM are becoming increasingly evident, and a new and more powerful class of techniques is emerging. For the first time in book form, Mesh Free Methods: Moving Beyond the Finite Element Method provides full, step-by-step details of techniques that can handle very effectively a variety of mechanics problems. The author systematically explores and establishes the theories, principles, and procedures that lead to mesh free methods. He shows that meshless methods not only accommodate complex problems in the mechanics of solids, structures, and fluids, but they do so with a significant reduction in pre-processing time. While they are not yet fully mature, mesh free methods promise to revolutionize engineering analysis. Filled with the new and unpublished results of the author's award-winning research team, this book is your key to unlocking the potential of these techniques, implementing them to solve real-world problems, and contributing to further advancements.
Few freeware FE programs offer the capabilities to include 3D finite deformation inelastic continuum analysis; those that do are typically expressed in tens of thousands of lines. This paper offers for the first time compact MATLAB scripts forming a complete finite deformation elasto–plastic FE program. The key modifications required to an infinitesimal FE program in order to include geometric non–linearity are described and the entire code given.
A new error estimator is presented which is not only reasonably accurate but whose evaluation is computationally so simple that it can be readily implemented in existing finite element codes. The estimator allows the global energy norm error to be well estimated and also gives a good evaluation of local errors. It can thus be combined with a full adaptive process of refinement or, more simply, provide guidance for mesh redesign which allows the user to obtain a desired accuracy with one or two trials. When combined with an automatic mesh generator a very efficient guidance process to analysis is available. Estimates other than the energy norm have successfully been applied giving, for instance, a predetermined accuracy of stresses.
A direct approach of coupling the element-free Galerkin method (EFGM) to both the finite element method (FEM) and the boundary element method (BEM) is applied to study dynamic soil-structure interactions. The structure and the soil body are assumed to be two-dimensional systems and discretized by EFG nodes, while the boundary of the soil region is modeled either by FEM or BEM to impose the boundary conditions in an easy way. Essential parameters in the EFG domain are chosen in a way where stable and reliable results at a relatively low cost are obtained. The accuracy and efficiency of the two new methodologies are compared to those of the conventional mesh-based approaches.