# Showing papers in "International Journal for Numerical Methods in Engineering in 1996"

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TL;DR: In this article, a consistent numerical solution procedure of the governing partial differential equations is presented, which is shown to be capable of properly simulating localization phenomena, and the introduction of higher-order deformation gradients in the constitutive model is demonstrated to be an adequate remedy to this deficiency of standard damage models.

Abstract: SUMMARY Conventional continuum damage descriptions of material degeneration suffer from loss of well-posedness beyond a certain level of accumulated damage. As a consequence, numerical solutions are obtained which are unacceptable from a physical point of view. The introduction of higher-order deformation gradients in the constitutive model is demonstrated to be an adequate remedy to this deficiency of standard damage models. A consistent numerical solution procedure of the governing partial differential equations is presented, which is shown to be capable of properly simulating localization phenomena.

1,207 citations

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TL;DR: In this article, the finite point method (FPM) is proposed for solving partial differential equations, which is based on a weighted least square interpolation of point data and point collocation for evaluating the approximation integrals.

Abstract: The paper presents a fully meshless procedure fo solving partial differential equations. The approach termed generically the ‘finite point method’ is based on a weighted least square interpolation of point data and point collocation for evaluating the approximation integrals. Some examples showing the accuracy of the method for solution of adjoint and non-self adjoint equations typical of convective-diffusive transport and also to the analysis of compressible fluid mechanics problem are presented.

809 citations

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TL;DR: In this paper, a new time-stepping method for simulating systems of rigid bodies is given which incorporates Coulomb friction and inelastic impacts and shocks, which does not need to identify explicitly impulsive forces.

Abstract: In this paper a new time-stepping method for simulating systems of rigid bodies is given which incorporates Coulomb friction and inelastic impacts and shocks. Unlike other methods which take an instantaneous point of view, this method does not need to identify explicitly impulsive forces. Instead, the treatment is similar to that of J. J. Moreau and Monteiro-Marques, except that the numerical formulation used here ensures that there is no inter-penetration of rigid bodies, unlike their velocity-based formulation. Numerical results are given for the method presented here for a spinning rod impacting a table in two dimensions, and a system of four balls colliding on a table in a fully three-dimensional way. These numerical results also show the practicality of the method, and convergence of the method as the step size becomes small.

644 citations

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TL;DR: The use of standard constitutive equations to model strong discontinuities (cracks, shear bands, slip lines, etc.) in solid mechanics analyzes is discussed in this paper.

Abstract: The paper addresses some fundamental aspects about the use of standard constitutive equations to model strong discontinuities (cracks, shear bands, slip lines, etc.) in solid mechanics analyzes. The strong discontinuity analysis is introduced as a basic tool to derive a general framework, in which different families of constitutive equations can be considered, that allows to extract some outstanding aspects of the intended analysis. In particular, a link between continuum and discrete approaches to the strain localization phenomena is obtained. Applications to standard continuum damage and elastoplastic constitutive equations are presented. Relevant aspects to be considered in the numerical simulation of the problem (tackled in Part 2 of the work) are also presented.

455 citations

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TL;DR: The element-free Galerkin method for dynamic crack propagation is described and applied to several problems as mentioned in this paper, which facilitates the modelling of growing crack problems because it does not require remeshing; the growth of the crack is modelled by extending its surfaces.

Abstract: The element-free Galerkin method for dynamic crack propagation is described and applied to several problems. This method is a gridless method, which facilitates the modelling of growing crack problems because it does not require remeshing; the growth of the crack is modelled by extending its surfaces. The essential feature of the method is the use of moving least-squares interpolants for the trial-and-test functions. In these interpolants, the dependent variable is obtained at any point by minimizing a weighted quadratic form involving the nodal variables within a small domain surrounding the point. The discrete equations are obtained by a Galerkin method. The procedures for modelling dynamic crack propagation based on dynamic stress intensity factors are also described.

372 citations

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TL;DR: A new simple explicit two-step method and a new family of predictor–corrector integration algorithms are developed for use in the solution of numerical responses of dynamic problems, avoiding solving simultaneous linear algebraic equations in each time step.

Abstract: A new simple explicit two-step method and a new family of predictor–corrector integration algorithms are developed for use in the solution of numerical responses of dynamic problems. The proposed integration methods avoid solving simultaneous linear algebraic equations in each time step, which is valid for arbitrary damping matrix and diagonal mass matrix frequently encountered in practical engineering dynamic systems. Accordingly, computational speeds of the new methods applied to large system analysis can be far higher than those of other popular methods. Accuracy, stability and numerical dissipation are investigated. Linear and nonlinear examples for verification and applications of the new methods to large-scale dynamic problems in railway engineering are given. The proposed methods can be used as fast and economical calculation tools for solving large-scale nonlinear dynamic problems in engineering.

337 citations

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TL;DR: In this paper, a finite element framework for the simulation of strong discontinuities, which belongs to the family of assumed enhanced strain methods, is presented, taking the standard linear triangle as the underlying element.

Abstract: On the basis of the strong discontinuity analysis of standard local stress–strain constitutive equations, a finite element framework for the simulation of strong discontinuities, which belongs to the family of assumed enhanced strain methods, is presented. Taking the standard linear triangle as the underlying element, an additional incompatible mode leads to the formulation of an enriched strain field which is shown to be able to appropriately capture strong discontinuities. The presented numerical simulations show that mesh size and mesh alignment dependencies can be completely removed.

285 citations

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TL;DR: In this article, a normalized smoothing function (NSF) algorithm was proposed to improve the accuracy of smooth particle hydrodynamics (SPH) impact computations for axisymmetric geometry.

Abstract: This paper presents a normalized smoothing function (NSF) algorithm that can improve the accuracy of smooth particle hydrodynamics (SPH) impact computations. It is presented specifically for axisymmetric geometry, but the principles also apply to plane strain and three-dimensional geometry. The approach consists of adjusting the standard smoothing functions for every node (and every cycle) such that the normal strain rates are computed exactly for conditions of constant strain rates (linear velocity distributions). This, in turn, generally improves accuracy for non-uniform strain rates. This can significantly improve the accuracy for free boundaries, for non-uniform arrangements of SPH nodes, and for small smoothing distances. A new smoothing function is also introduced. The NSF algorithm is shown to provide improved accuracy for a series of cylinder impact examples that includes two different smoothing functions and two different smoothing distances.

257 citations

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TL;DR: This paper introduces a new algorithm called whisker weaving for constructing unstructured, all-hexahedral finite element meshes based on the Spatial Twist Continuum (STC), a global interpretation of the geometric dual of an all- hexahedral mesh.

Abstract: This paper introduces a new algorithm called whisker weaving for constructing unstructured, all-hexahedral finite element meshes. Whisker weaving is based on the Spatial Twist Continuum (STC), a global interpretation of the geometric dual of an all-hexahedral mesh. Whisker weaving begins with a closed, all-quadrilateral surface mesh bounding a solid geometry, then constructs hexahedral element connectivity advancing into the solid. The result of the whisker weaving algorithm is a complete representation of hex mesh connectivity only: Actual mesh node locations are determined afterwards.
The basic step of whisker weaving is to form a hexahedral element by crossing or intersecting dual entities. This operation, combined with seaming or joining operations in dual space, is sufficient to mesh simple block problems. When meshing more complex geometries, certain other dual entities appear such as blind chords, merged sheets, and self-intersecting chords. Occasionally specific types of invalid connectivity arise. These are detected by a general method based on repeated STC edges. This leads into a strategy for resolving some cases of invalidities immediately.
The whisker weaving implementation has so far been successful at generating meshes for simple block-type geometries and for some non-block geometries. Mesh sizes are currently limited to a few hundred elements. While the size and complexity of meshes generated by whisker weaving are currently limited, the algorithm shows promise for extension to much more general problems.

256 citations

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TL;DR: In this article, the authors deal with the continuum formulation of finite strain viscoelasticity and provide its numerical simulation with the finite element method with the main thrust of this paper on the computational side is to show the meaningful time-dependent behaviour and the general applicability of the three-dimensional constitutive model.

Abstract: This article deals with the continuum formulation of finite strain viscoelasticity and provides its numerical simulation with the finite element method. In particular, elastomeric solids which are of essential engineering interest are discussed. In order to simulate the significant different bulk/shear-response of polymeric media the deformation is decomposed into volumetric elastic and isochoric viscoelastic parts. The constitutive equations are presented within the context of internal variable models and a Lagrangian kinematical description is adopted throughout. For sufficiently slow processes the material responds in a rubbery elastic manner which is assumed to be modelled with an Ogden-type strain energy function well-known from rubber elasticity. The stresses and the symmetric consistent tangent moduli are briefly discussed along with a second-order approximation of the constitutive rate equation. The main thrust of this paper on the computational side is to show the meaningful time-dependent behaviour and the general applicability of the three-dimensional constitutive model. By applying assumed enhanced strain elements which are well-suited for (nearly) incompressible problems three representative numerical examples illustrate relaxation and creeping phenomena at large strains and the equilibrium finite elastic response, which is asymptotically obtained.

247 citations

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TL;DR: In this paper, improved algorithms are proposed for a gradient plasticity theory in which the Laplacian of an invariant plastic strain measure enters the yield function, and the type of finite elements that can be used within the format of gradient-dependent plasticity.

Abstract: SUMMARY Improved algorithms are proposed for a gradient plasticity theory in which the Laplacian of an invariant plastic strain measure enters the yield function. Particular attention is given to the type of finite elements that can be used within the format of gradient-dependent plasticity. Assuming a weak satisfaction of the yield function, mixed finite elements are developed, in which the invariant plastic strain measure and the displacements are discretized. Two families of finite elements are developed: one in which the invariant plastic strain measure is interpolated using C '-continuous polynomials, and one in which penalty-enhanced C'continuous interpolants are used. The performance of both families of finite elements is assessed numerically in one-dimensional and two-dimensional boundary value problems. The regularizing effect of the used gradient enhancement in computations of elastoplastic solids is demonstrated, both for mesh refinement and for the directional bias of the grid lines.

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TL;DR: In this paper, a multilayered composite plate element is proposed that includes both the zig-zag distribution along the thickness co-ordinate of the in-plane displacements and the interlaminar continuity (equilibrium) for the transverse shear stresses.

Abstract: SUMMARY Concerning composites plate theories and FEM (Finite Element Method) applications this paper presents some multilayered plate elements which meet computational requirements and include both the zig-zag distribution along the thickness co-ordinate of the in-plane displacements and the interlaminar continuity (equilibrium) for the transverse shear stresses. This is viewed as the extension to multilayered structures of well-known Co Reissner-Mindlin finite plate elements. Two different fields along the plate thickness co-ordinate are assumed for the transverse shear stresses and for the displacements, respectively. In order to eliminate stress unknowns, reference is made to a Reissner mixed variational theorem. Sample tests have shown that the proposed elements, named RMZC, numerically work as the standard Reissner-Mindlin ones. Furthermore, comparisons with other results related to available higher-order shear deformation theories and to three-dimensional solutions have demonstrated the good performance of the RMZC elements. Major portions of aerospace structures, as well as automotive and ship vehicles consist of flat and curved panels that are used as primary load-carrying components. Due to their obvious advantages, such as critical strength/stiffness-to-weight ratios, an increasing number of these panels are made of laminated composite material. This has led to extensive research activities in the mechanical properties, loading behaviour, structural modelling, and failure assessment of multilayered composite structures. Due to the geometry of laminated structural components, two-dimensional approaches have been extensively used to trace their response. The classical Kirchhoff's plate theory (CLT, Classical Lamination Theory) has revealed its limits when applied to thick panels with high orthotropic ratio.' - The shear deformation theories of Reissner-Mindlin-type (FSDT, First Shear Deformation Theories), even though, they are quite acceptable to study global response of high shear deformable thick composite structures, are not adequate for forecasting local stress-strain characteristics. In fact, some representative problems, exact three-dimensional have shown the failure of FSDT both to fulfill the interlaminar transverse shear stresses continuity at each interface and to describe the so-called zig-zag form' of the

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TL;DR: The Galerkin method enriched with residual-free bubbles is considered for approximating the solution of the Helmholtz equation as discussed by the authors, and two-dimensional tests demonstrate the improvement over the standard GAs and the GAs using piecewise bilinear interpolations.

Abstract: The Galerkin method enriched with residual-free bubbles is considered for approximating the solution of the Helmholtz equation. Two-dimensional tests demonstrate the improvement over the standard Galerkin method and the Galerkin-least-squares method using piecewise bilinear interpolations.

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TL;DR: In this paper, exact evaluations of various finite integrals whose integrands involve products of Daubechies' compactly supported wavelets and their derivatives and/or integrals are given.

Abstract: This paper describes exact evaluations of various finite integrals whose integrands involve products of Daubechies' compactly supported wavelets and their derivatives and/or integrals. These finite integrals play an essential role in the wavelet-Galerkin approximation of differential or integral equations on a bounded interval.

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TL;DR: It was concluded that the voxel conversion method in combination with these solving methods not only provides a powerful tool to analyse structures that can not be analysed in another way, but also that this approach can be competitive with traditional meshing and solving techniques.

Abstract: SUMMARY FE-models for structural solid mechanics analyses can be readily generated from computer images via a ‘voxel conversion’ method, whereby voxels in a two- or three-dimensional computer image are directly translated to elements in a FE-model. The fact that all elements thus generated are the same creates the possibilities for fast solution algorithms that can compensate for a large number of elements. The solving methods described in this paper are based on an iterative solving algorithm in combination with a uniqueelement Element-by-Element (EBE) or with a newly developed Row-by-Row (RBR) matrix-vector multiplication strategy. With these methods it is possible to solve FE-models on the order of lo5 3-D brick elements on a workstation and on the order of lo6 elements on a Cray computer. The methods are demonstrated for the Boussinesq problem and for FE-models that represent a porous trabecular bone structure. The results show that the RBR method can be 3.2 times faster than the EBE method. It was concluded that the voxel conversion method in combination with these solving methods not only provides a powerful tool to analyse structures that can not be analysed in another way, but also that this approach can be competitive with traditional meshing and solving techniques.

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TL;DR: In this article, the authors present a new stress update algorithm for large-strain rate-independent single-crystal plasticity. But this algorithm is based on the exponential map exp: sl(3) SL(3), which is a special linear group of unimodular plastic deformation maps.

Abstract: This paper presents a new stress update algorithm for large-strain rate-independent single-crystal plasticity. The theoretical frame is the well-established continuum slip theory based on the multiplicative decomposition of the deformation gradient into elastic and plastic parts. A distinct feature of the present formulation is the introduction and computational exploitation of a particularly simple hyperelastic stress response function based on a further multiplicative decomposition of the elastic deformation gradient into spherical and unimodular parts, resulting in a very convenient representation of the Schmid resolved shear stresses on the crystallographic slip systems in terms of a simple inner product of Eulerian vectors. The key contribution of this paper is an algorithmic formulation of the exponential map exp: sl(3) SL(3) for updating the special linear group SL(3) of unimodular plastic deformation maps. This update preserves exactly the plastic incompressibility condition of the anisotropic plasticity model under consideration. The resulting fully implicit stress update algorithm treats the possibly redundant constraints of single-crystal plasticity by means of an active set search. It exploits intrinsically the simple representation of the Schmid stresses by formulating the return algorithm and the associated consistent elastoplastic moduli in terms of Eulerian vectors updates. The performance of the proposed algorithm is demonstrated by means of a representative numerical example.

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TL;DR: In this paper, it is shown that for one-dimensional Hadamard finite-part integrals, the even part of the integrand has a Holder-continuous first derivative, while the odd part is discontinuous.

Abstract: Hypersingular integrals are guaranteed to exist at a point x only if the density function f in the integrand satises certain conditions in a neighbourhood of x. It is well known that a su- cient condition is that f has a Holder-continuous rst derivative. This is a stringent condition, especially when it is incorporated into boundary-element methods for solving hypersingular in- tegral equations. This paper is concerned with nding weaker conditions for the existence of one-dimensional Hadamard nite-part integrals: it is shown that it is sucien t for the even part of f (with respect to x) to have a Holder-continuous rst derivative { the odd part is al- lowed to be discontinuous. A similar condition is obtained for Cauchy principal-value integrals. These simple results have non-trivial consequences. They are applied to the calculation of the tangential derivative of a single-layer potential and to the normal derivative of a double-layer potential. Particular attention is paid to discontinuous densities and to discontinuous boundary conditions. Also, despite the weaker sucien t conditions, it is rearmed that, for hypersingular integral equations, collocation at a point x at the junction between two standard conforming boundary elements is not permissible, theoretically. Various modications to the denition of nite-part integral are explored.

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TL;DR: In this article, a model describing the dynamical behavior of a saturated binary porous medium including both geometrical and material non-linearities is presented and solved by using the finite element method.

Abstract: Based on the theory of porous media (mixture theories extended by the concept of volume fractions), a model describing the dynamical behaviour of a saturated binary porous medium is presented including both geometrical and material non-linearities. Transformed toward a weak formulation, the model equations are solved by use of the finite element method. Applications of the model range from one-dimensional linear problems to two-dimensional problems including the full dynamics and non-linearities.

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Mines ParisTech

^{1}TL;DR: A least-squares-type algorithm is suggested for the unconstrained optimization method (based on external penalty) for which it can reduce to calculations which are equivalent to the derivative calculations of steady-state processes and to evolution equations.

Abstract: We suggest a shape optimization method for a non-linear and non-steady-state metal forming problem. It consists in optimizing the initial shape of the part as well as the shape of the preform tool during a two-step forging operation, for which the shape of the second operation is known. Shapes are described using spline functions and optimal parameter values of the splines are searched in order to produce, at the end of the forging sequence, a part with a prescribed geometric accuracy, optimal metallurgical properties and for a minimal production cost. The finite element method, including numerous remeshing operations, is used for the simulation of the process. We suggest using a least-squares-type algorithm for the unconstrained optimization method (based on external penalty) for which we describe the calculation of the derivatives of the objective function. We show that it can reduce to calculations which are equivalent to the derivative calculations of steady-state processes and to evolution equations. Therefore, the computational cost of such an optimization is quite reasonable, even for complex forging processes. Lastly, in order to reduce the errors due to the numerous remeshings during the simulation, we introduce error estimation and adaptive remeshing methods with respect to the calculation of derivatives.

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TL;DR: In this work a population of binary strings or ‘chromosomes’ are used, which represent the coded truss design variables, a ‘fitness’ as a ranking measure of the adaptability to the environment, selection criteria and mechanical natural operators such as crossover and mutation are used to improve the population.

Abstract: Genetic algorithms, a search technique which combines Darwinian ‘survival-of-the-fittest’ with randomized well structured information, is applied to the problems of real-world truss optimization. In this work a population of binary strings or ‘chromosomes’, which represent the coded truss design variables, a ‘fitness’ as a ranking measure of the adaptability to the environment, selection criteria and mechanical natural operators such as crossover and mutation are used to improve the population, so that over the generations the genetic algorithm gets better and better and at the end of the convergence, a ‘rebirth’ of the population is used to improve the usual process.
An overview of the genetic algorithm will be described, continuing the rebirth effect; then, the chromosome representation of trusses is exposed. Afterwards, the objective scalar function is defined taking into account that it seems reasonable in real world to optimize trusses in minimum weight trying, at the same time, to use the minimum number of cross-section types obtained from the market. It also seems reasonable to have the possibility to change the shape of the conceptual design, moving some joints. To simulate nearly real conditions, several load cases, constraints in the elastic joint displacements, ultimate tensile and elastic and plastic buckling in the bars have been taken into account. A hyperstatic 10 bars truss is subjected to a deep analysis in different situations in order to evaluate with other authors when possible as truss optimization with two criteria and buckling effect has not been found in specialized literature. A 160-bar transmission tower is also optimized.

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TL;DR: In this paper, the authors used a DYNA3D finite element analysis to find elastic moduli as a function of the reduced volume, and a homogenization method was developed to find plastic moduli and yield stress as functions of density.

Abstract: The topological optimization of components to maximize crash energy absorption for a given volume is considered. The crash analysis is performed using a DYNA3D finite element analysis. The original solid elements are replaced by ones with holes, the hole size being characterized by a so-called density (measure of the reduced volume). A homogenization method is used to find elastic moduli as a function of this density. Simpler approximations were developed to find plastic moduli and yield stress as functions of density.
Optimality criteria were derived from an optimization statement using densities as the design variables. A resizing algorithm was constructed so that the optimality criteria are approximately satisfied. A novel feature is the introduction of an objective function based on strain energies weighted at specified times. Each different choice of weighting factors leads to a different structure, allowing a range of design possibilities to be explored.
The method was applied to an automotive body rear rail. The original design and a new design of equal volume with holes were compared for energy absorption.

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TL;DR: In this article, a time-discontinuous Galerkin finite element method for structural dynamic problems is proposed, by which both displacements and velocities are approximated as piecewise linear functions in the time domain and may be discontinuous at the discrete time levels.

Abstract: This paper studies a time-discontinuous Galerkin finite element method for structural dynamic problems, by which both displacements and velocities are approximated as piecewise linear functions in the time domain and may be discontinuous at the discrete time levels A new iterative solution algorithm which involves only one factorization for each fixed time step size and a few iterations at each step is presented for solving the resulted system of coupled equations By using the jumps of the displacements and the velocities in the total energy norm as error indicators, an adaptive time-stepping procedure for selecting the proper time step size is described Numerical examples including both single-DOF and multi-DOF problems are used to illustrate the performance of these algorithms Comparisons with the exact results and/or the results by the Newmark integration scheme are given It is shown that the time-discontinuous Galerkin finite element method discussed in this study possesses good accuracy (third order) and stability properties, its numerical implementation is not difficult, and the higher computational cost needed in each time step is compensated by use of a larger time step size

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TL;DR: In this paper, a consistent infinitesimal finite-element cell method was developed for the three-dimensional vector wave equation, which does not require a fundamental solution and incorporates interfaces extending from the structure medium interface to infinity compatible with similarity without any additional computational effort.

Abstract: To calculate the unit-impulse response matrix of an unbounded medium for use in a time-domain analysis of unbounded medium–structure interaction, the consistent infinitesimal finite-element cell method is developed for the three-dimensional vector wave equation. This is a boundary finite-element procedure. The discretization is only performed on the structure–medium interface, yielding a reduction of the spatial dimension by 1. The procedure is rigorous in the radial direction and exact in the finite-element sense in the circumferential directions. In contrast to the boundary-element procedure, the consistent infinitesimal finite-element cell method does not require a fundamental solution and incorporates interfaces extending from the structure–medium interface to infinity compatible with similarity without any additional computational effort. A general anisotropic material can be processed. The derivation is based on the finite-element formulation and on similarity.

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TL;DR: Methods of calculating angles, projecting elements, and detecting collisions between paving boundaries, for general surfaces are presented, and advances in the use of scalar sizing functions can be used to better approximate internal mesh density from boundary densities and surface characteristics.

Abstract: This paper discusses the extension of the paving algorithm for all quadrilateral mesh generation to arbitrary three-dimensional trimmed surfaces. Methods of calculating angles, projecting elements, and detecting collisions between paving boundaries, for general surfaces are presented. Extensions of the smoothing algorithms for three dimensions are set forth. Advances in the use of scalar sizing functions are presented. These functions can be used to better approximate internal mesh density from boundary densities and surface characteristics.

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TL;DR: In this article, a nonlocal concept based on microcrack interactions is discussed, its implementation in a smeared cracking finite element code for concrete is presented, numerical studies are reported, and comparisons with experimental results are made.

Abstract: SUMMARY A recently proposed new nonlocal concept based on microcrack interactions is discussed, its implementation in a smeared cracking finite element code for concrete is presented, numerical studies are reported, and comparisons with experimental results are made. The nonlocality is not merely a mathematical device to prevent excessive spurious localization into a zone of zero volume but is a necessary physical consequence of microcrack interactions. Since the constitutive law itself is strictly local, the new nonlocal concept can be combined with any type of constitutive law for strain-softening nonlocal damage, which is here chosen to be the micro plane model. A simple method is formulated to approximately identify the material parameters in the model from the basic characteristics of concrete such as the tensile strength, fracture energy and maximum aggregate size. The results of finite element analysis are shown to be mesh insensitive, and good convergence is obtained. Cracking damage is found to localize into a volume whose size and shape depend on the macroscopic concrete properties as well as the current stress-strain state. Although the damage is considered to be tensile on the microlevel, due solely to mode I microcracks, the new non local model can describe well not only mode I fracture tests but also complex shear-dominated and mixed-mode types of failure such a diagonal shear, and can do so for the same values of material parameters (which was not the case for previous nonlocal models). Most importantly, the new nonlocal model can correctly capture the size effect of quasibrittle fracture, in approximate agreement with Bazanfs size effect law.

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Mines ParisTech

^{1}TL;DR: In this article, the shape optimization of hot axisymmetrical forging is studied. But the main feature of this work is the analytical calculations of the derivatives of the objective function for a non-linear, non-steady-state problem with large deformations.

Abstract: This paper is the second part of a two-part article about shape optimization of metal forming processes. This part is focused on numerical applications of the optimization method which has been described in the first paper. The main feature of this work is the analytical calculations of the derivatives of the objective function for a non-linear, non-steady-state problem with large deformations. The calculations are based on the differentiation of the discrete objective function and on the differentiation of the discrete equations of the forging problem. Our aim here is to show the feasibility and the efficiency of such a method with numerical examples. We recall the formulation and the resolution of the direct problem of hot axisymmetrical forging. Then, a first type of shape optimization problem is considered: the optimization of the shape of the initial part for a one-step forging operation. Two academic problems allow for checking the accuracy of the analytical derivatives, and for studying the convergence rate of the optimization procedure. Both constrained and unconstrained problems are considered. Afterwards, a second type of inverse problem of design is considered: the shape optimization of the preforming tool, for a two-step forging process. A satisfactory shape is obtained after few iterations of the optimization procedure.

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TL;DR: In this paper, a method of generating general tetrahedral meshes suitable for use in viscous flow simulations is proposed, which consists of the initial generation of a number of unstructured layers of highly stretched elements, in the vicinity of solid walls, followed by the discretisation of the remainder of the domain, by a standard advancing front procedure.

Abstract: A method of generating general tetrahedral meshes suitable for use in viscous flow simulations is proposed. The approach which is followed consists of the initial generation of a number of unstructured layers of highly stretched elements, in the vicinity of solid walls, followed by the discretisation of the remainder of the domain, by a standard advancing front procedure. The numerical performance of the proposed methodology is demonstrated by the generation of meshes suitable for viscous flow analysis over a number of three-dimensional aerodynamic configurations of current practical interest.