3D NURBS‐enhanced finite element method (NEFEM)

This paper presents the extension of the recently proposed NURBS‐enhanced finite element method (NEFEM) to 3D domains. NEFEM is able to exactly represent the geometry of the computational domain by means of its CAD boundary representation with non‐uniform rational B‐splines (NURBS) surfaces. Specific strategies for interpolation and numerical integration are presented for those elements affected by the NURBS boundary representation. For elements not intersecting the boundary, a standard finite element rationale is used, preserving the efficiency of the classical FEM. In 3D NEFEM special attention must be paid to geometric issues that are easily treated in the 2D implementation. Several numerical examples show the performance and benefits of NEFEM compared with standard isoparametric or cartesian finite elements. NEFEM is a powerful strategy to efficiently treat curved boundaries and it avoids excessive mesh refinement to capture small geometric features. Copyright © 2011 John Wiley & Sons, Ltd.     

A distortion measure to validate and generate curved high‐order meshes on CAD surfaces with independence of parameterization

A framework to validate and generate curved nodal high‐order meshes on Computer‐Aided Design (CAD) surfaces is presented. The proposed framework is of major interest to generate meshes suitable for thin‐shell and 3D finite element analysis with unstructured high‐order methods. First, we define a distortion (quality) measure for high‐order meshes on parameterized surfaces that we prove to be independent of the surface parameterization. Second, we derive a smoothing and untangling procedure based on the minimization of a regularization of the proposed distortion measure. The minimization is performed in terms of the parametric coordinates of the nodes to enforce that the nodes slide on the surfaces. Moreover, the proposed algorithm repairs invalid curved meshes (untangling), deals with arbitrary polynomial degrees (high‐order), and handles with low‐quality CAD parameterizations (independence of parameterization). Third, we use the optimization procedure to generate curved nodal high‐order surface meshes by means of an a posteriori approach. Given a linear mesh, we increase the polynomial degree of the elements, curve them to match the geometry, and optimize the location of the nodes to ensure mesh validity. Finally, we present several examples to demonstrate the features of the optimization procedure, and to illustrate the surface mesh generation process. Copyright © 2015 John Wiley & Sons, Ltd.     

Automatic remeshing in 3‐D analysis of forming processes

Finite element modelling, employing updated Lagrangian techniques, is used extensively in the design and analysis of bulk forming processes. However, the full 3‐D capability has not seen widespread use in the automotive, aerospace, and, related industries due to, among other reasons, the need for remeshing, or, representation of the workpiece with a new finite element mesh as the analysis progresses. Automating the remeshing procedure of the deformed workpiece geometry would reduce the time required for a 3‐D analysis by several orders of magnitude. This paper discusses an algorithm for generating a new mesh to represent the deformed workpiece geometry during the analysis. The procedure is used to perform a 3‐D analysis of a valve forging problem. Copyright © 1999 John Wiley & Sons, Ltd.     

Immersed stress method for fluid–structure interaction using anisotropic mesh adaptation

This paper presents advancements toward a monolithic solution procedure and anisotropic mesh adaptation for the numerical solution of fluid–structure interaction with complex geometry. First, a new stabilized three‐field stress, velocity, and pressure finite element formulation is presented for modeling the interaction between the fluid (laminar or turbulent) and the rigid body. The presence of the structure will be taken into account by means of an extra stress in the Navier–Stokes equations. The system is solved using a finite element variational multiscale method. We combine this method with anisotropic mesh adaptation to ensure an accurate capturing of the discontinuities at the fluid–solid interface. We assess the behavior and accuracy of the proposed formulation in the simulation of 2D and 3D time‐dependent numerical examples such as the flow past a circular cylinder and turbulent flows behind an immersed helicopter in a forward flight. Copyright © 2013 John Wiley & Sons, Ltd.     

Finite cover method with mortar elements for elastoplasticity problems

Finite cover method (FCM) is extended to elastoplasticity problems. The FCM, which was originally developed under the name of manifold method, has recently been recognized as one of the generalized versions of finite element methods (FEM). Since the mesh for the FCM can be regular and squared regardless of the geometry of structures to be analyzed, structural analysts are released from a burdensome task of generating meshes conforming to physical boundaries. Numerical experiments are carried out to assess the performance of the FCM with such discretization in elastoplasticity problems. Particularly to achieve this accurately, the so-called mortar elements are introduced to impose displacement boundary conditions on the essential boundaries, and displacement compatibility conditions on material interfaces of two-phase materials or on joint surfaces between mutually incompatible meshes. The validity of the mortar approximation is also demonstrated in the elastic-plastic FCM.     

Numerical simulations of strain localization in inelastic solids using mesh‐free methods

In this paper, a comprehensive account on using mesh‐free methods to simulate strain localization in inelastic solids is presented. Using an explicit displacement‐based formulation in mesh‐free computations, high‐resolution shear‐band formations are obtained in both two‐dimensional (2‐D) and three‐dimensional (3‐D) simulations without recourse to any mixed formulation, discontinuous/incompatible element or special mesh design. The numerical solutions obtained here are insensitive to the orientation of the particle distributions if the local particle distribution is quasi‐uniform, which, to a large extent, relieves the mesh alignment sensitivity that finite element methods suffer. Moreover, a simple h‐adaptivity procedure is implemented in the explicit calculation, and by utilizing a mesh‐free hierarchical partition of unity a spectral (wavelet) adaptivity procedure is developed to seek high‐resolution shear‐band formations. Moreover, the phenomenon of multiple shear band and mode switching are observed in numerical computations with a relatively coarse particle distribution in contrast to the costly fine‐scale finite element simulations. Copyright © 2000 John Wiley & Sons, Ltd.     

3D‐based hp‐adaptive first‐order shell finite element for modelling and analysis of complex structures—Part 1: The model and the approximation

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 this paper, is to provide non‐standard information on the original parts of the element algorithm. In order to do that, we present the definition of shape functions necessary for p‐adaptivity, as well as the procedure for imposing constraints corresponding to the lack of elongation of the straight lines perpendicular to the shell mid‐surface, which is the procedure necessary for q‐adaptivity. The 3D version of constrained approximation presented next is the basis for h‐adaptivity of the element. The second part of our research, devoted to methodology and results of the numerical research on application of the element to various plate and shell problems, are described in the second part of this paper. Copyright © 2006 John Wiley & Sons, Ltd.     

Mesh independent analysis of shell‐like structures

Mesh independent analysis is motivated by the desire to use accurate geometric models represented as equations rather than approximated by a mesh. The trial and test functions are approximated or interpolated on a background mesh that is independent of the geometry. This background mesh is easy to generate because it does not have to conform to the geometry. Essential boundary conditions can be applied using the implicit boundary method where the trial and test functions are constructed utilizing approximate step functions such that the boundary conditions are guaranteed to be satisfied. This approach has been demonstrated for two‐dimensional (2D) and three‐dimensional (3D) structural analysis and is extended in this paper to model shell‐like structures. The background mesh consists of 3D elements that use uniform B‐spline approximations, and the shell geometry is assumed to be defined as parametric surfaces to allow arbitrarily complex shell‐like structures to be modeled. Several benchmark problems are used to study the validity of these 3D B‐spline shell elements. Copyright © 2012 John Wiley & Sons, Ltd.     

Initialization of high‐order accuracy immersed interface CFD solvers using complex CAD geometry

This study concerns the development of a numerical methodology for initializing immersed interface‐based CFD solvers for using complex computer‐aided design (CAD) geometry. CFD solvers with higher‐order discretization stencils require larger stencil widths, which become problematic in regions of space where insufficient mesh resolution is available. This problem becomes especially challenging when convoluted triangulated surface meshes generated from complex solid models are used to initialize the cut‐cells. A pragmatic balance between desired local geometry resolution and numerical accuracy is often required to find a practical solution. Here, a robust iterative fill algorithm is presented that balances geometry resolution with numerical accuracy (via stencil size). Several examples are presented to illustrate the use of this initialization procedure that employs both the original CAD generated triangulated surface mesh, along with a level set representation of the surface to initialize cut‐cells and boundary proximity measures for creation of CFD stencils. Convergence error analysis of surface area and enclosed volumes is first presented to show the effects of fill on the geometry as a function of desired stencil size and grid resolution. The algorithm is then applied to geometrically complex problems using large eddy simulation. Two problems are considered. The first is flow around the Eiffel Tower. The second is a combustion swirler in the context of a design problem. Copyright © 2016 John Wiley & Sons, Ltd.     

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.     

Non‐reflecting boundary conditions for atomistic,continuum and coupled atomistic/continuum simulations

We present a method to numerically calculate a non‐reflecting boundary condition which is applicable to atomistic, continuum and coupled multiscale atomistic/continuum simulations. The method is based on the assumption that the forces near the domain boundary can be well represented as a linear function of the displacements, and utilizes standard Laplace and Fourier transform techniques to eliminate the unnecessary degrees of freedom. The eliminated degrees of freedom are accounted for in a time‐history kernel that can be calculated for arbitrary crystal lattices and interatomic potentials, or regular finite element meshes using an automated numerical procedure. The new theoretical developments presented in this work allow the application of the method to non‐nearest neighbour atomic interactions; it is also demonstrated that the identical procedure can be used for finite element and mesh‐free simulations. We illustrate the effectiveness of the method on a one‐dimensional model problem, and calculate the time‐history kernel for FCC gold using the embedded atom method (EAM). Copyright © 2005 John Wiley & Sons, Ltd.     

Phase‐field material point method for brittle fracture

The material point method for the analysis of deformable bodies is revisited and originally upgraded to simulate crack propagation in brittle media. In this setting, phase‐field modelling is introduced to resolve the crack path geometry. Following a particle in cell approach, the coupled continuum/phase‐field governing equations are defined at a set of material points and interpolated at the nodal points of an Eulerian, ie, non‐evolving, mesh. The accuracy of the simulated crack path is thus decoupled from the quality of the underlying finite element mesh and relieved from corresponding mesh‐distortion errors. A staggered incremental procedure is implemented for the solution of the discrete coupled governing equations of the phase‐field brittle fracture problem. The proposed method is verified through a series of benchmark tests while comparisons are made between the proposed scheme, the corresponding finite element implementation, and experimental results.     

Finite cover method for linear and non‐linear analyses of heterogeneous solids

We introduce the finite cover method (FCM) as a generalization of the finite element method (FEM) and extend it to analyse the linear and non‐linear mechanical behaviour of heterogeneous solids and structures. The name ‘FCM’ is actually an alias for the manifold method (MM) and the basic idea of the method has already been established for linear analyses of structures with homogeneous materials. After reviewing the concept of physical and mathematical covers for approximating functions in the FCM, we present the formulation for the static equilibrium state of a structure with arbitrary physical boundaries including material interfaces. The problem essentially involves the discontinuities in strains, and possibly has the discontinuities in displacement caused by interfacial debonding or rupture of material interfaces. We simulate such non‐linear mechanical behaviour after presenting simple numerical examples that demonstrate the equivalence between the approximation capabilities of the FCM and those of the FEM. Copyright © 2003 John Wiley & Sons, Ltd.     

Failure analysis of quasi‐brittle materials involving multiple mechanisms on fractured surfaces

We present a variational formulation for the quasi‐static boundary value problem of a structure with quasi‐brittle materials, involving (i) unknown states of contact, (ii) deformation‐dependent frictional forces, (iii) crack opening and closing with cohesive traction, and (iv) configuration change due to the initiation and the evolution of cracks, and propose a new finite cover method (FCM) capable of reflecting those multiple mechanisms in the failure analysis. The cover‐division strategy is also introduced to judge the generation of cracks, and to locate and orient them within the framework of the FCM. A relevant numerical algorithm is designed to be consistent with the mathematical representation of the multiple mechanisms. Several numerical examples are presented to validate the proposed method and to demonstrate its promise and potential for evaluating the ultimate strength of quasi‐brittle materials. Copyright © 2006 John Wiley & Sons, Ltd.     

Crack growth analysis using mesh superposition technique and X‐FEM

In this paper, a new approach to crack propagation analysis for large scale or complicated geometry structures is presented. That is the connection of mesh superposition technique and the extended finite element method. The former is a technique that increases the accuracy of analysis locally by superimposing additional mesh of higher resolution onto the mesh that represents the rough deformation of the structure. In this technique, the boundaries and nodes of both meshes do not have to coincide with each other. In the latter technique, the discontinuity across the crack segment and singularity around the crack tips are represented in the approximation by enriching the nodal degrees using partition of unity condition. This technique does not require meshes to conform to the crack geometry and this enables crack propagation procedure with no re‐meshing process by connecting both techniques, it becomes possible to analyse crack growth models of large scale and complicated shape with highly flexible modelling and no remeshing process. By using this approach, some numerical examples are analysed and appropriate results are obtained. Copyright © 2007 John Wiley & Sons, Ltd.     

Automatic Meshing and Remeshing in the Simultaneous Engineering Context

This paper presents a method to integrate in a better way the finite element method in the CAD/CAM process for two-dimensional problems, through efficient and automatic meshing and remeshing procedures. During the design step, the lack of integration between geometric modeling and numerical analysis remains a crucial problem and it still tends to restrain the use of finite element methods to a small number of engineers. Here we tackle the problem of the automatic remeshing of an object in the context of minor changes in its geometry and topology without restarting the mesh generation from the beginning. We have developed a mesh generator that is able to adapt a previous mesh, through two complementary strategies (for 2D cases) to a new geometry without destroying the whole initial discretization. We also present the possible extension of these concepts to three-dimensional problems.     

Three‐dimensional immersed finite element methods for electric field simulation in composite materials

This paper presents two immersed finite element (IFE) methods for solving the elliptic interface problem arising from electric field simulation in composite materials. The meshes used in these IFE methods can be independent of the interface geometry and position; therefore, if desired, a structured mesh such as a Cartesian mesh can be used in an IFE method to simulate 3‐D electric field in a domain with non‐trivial interfaces separating different materials. Numerical examples are provided to demonstrate that the accuracies of these IFE methods are comparable to the standard linear finite element method with unstructured body‐fit mesh. Copyright © 2005 John Wiley & Sons, Ltd.     

The 3‐D BEM implementation of a numerical Green's function for fracture mechanics applications

The use of Green's functions has been considered a powerful technique in the solution of fracture mechanics problems by the boundary element method (BEM). Closed‐form expressions for Green's function components, however, have only been available for few simple 2‐D crack geometry applications and require complex variable theory. The present authors have recently introduced an alternative numerical procedure to compute the Green's function components that produced BEM results for 2‐D general geometry multiple crack problems, including static and dynamic applications. This technique is not restricted to 2‐D problems and the computational aspects of the 3‐D implementation of the numerical Green's function approach are now discussed, including examples. Copyright © 2000 John Wiley & Sons, Ltd.     

Finite cover method for progressive failure with cohesive zone fracture in heterogeneous solids and structures

The finite cover method (FCM), which is a cover-based generalized finite element method, is extended for analyses of progressive failure processes involving cohesive zone fracture, starting from an interface debonding and evolving toward one of the constituents of heterogeneous solids and structures. Assuming that the constituents fail according to the maximum principal stress, we are able to represent the evolution of the resulting failure surfaces of discontinuity independent of mesh alignment owing to the distinctive features of the FCM. Also, interface elements with Lagrange multipliers are introduced to impose compatibility conditions on the material interface so that debonding is judged by the multipliers. Representative numerical examples demonstrate the capability of the proposed method in tracing the smooth transition of crack paths from interfacial to internal failure, and vice versa.     

Influence functions and goal‐oriented error estimation for finite element analysis of shell structures

In this paper, we first present a consistent procedure to establish influence functions for the finite element analysis of shell structures, where the influence function can be for any linear quantity of engineering interest. We then design some goal‐oriented error measures that take into account the cancellation effect of errors over the domain to overcome the issue of over‐estimation. These error measures include the error due to the approximation in the geometry of the shell structure. In the calculation of the influence functions we also consider the asymptotic behaviour of shells as the thickness approaches zero. Although our procedures are general and can be applied to any shell formulation, we focus on MITC finite element shell discretizations. In our numerical results, influence functions are shown for some shell test problems, and the proposed goal‐oriented error estimation procedure shows good effectivity indices. Copyright © 2005 John Wiley & Sons, Ltd.