Object‐oriented finite element analysis of thermo‐hydro‐mechanical (THM) problems in porous media

The design, implementation and application of a concept for object‐oriented in finite element analysis of multi‐field problems is presented in this paper. The basic idea of this concept is that the underlying governing equations of porous media mechanics can be classified into different types of partial differential equations (PDEs). In principle, similar types of PDEs for diverse physical problems differ only in material coefficients. Local element matrices and vectors arising from the finite element discretization of the PDEs are categorized into several types, regardless of which physical problem they belong to (i.e. fluid flow, mass and heat transport or deformation processes). Element (ELE) objects are introduced to carry out the local assembly of the algebraic equations. The object‐orientation includes a strict encapsulation of geometrical (GEO), topological (MSH), process‐related (FEM) data and methods of element objects. Geometric entities of an element such as nodes, edges, faces and neighbours are abstracted into corresponding geometric element objects (ELE–GEO). The relationships among these geometric entities form the topology of element meshes (ELE–MSH). Finite element objects (ELE–FEM) are presented for the local element calculations, in which each classification type of the matrices and vectors is computed by a unique function. These element functions are able to deal with different element types (lines, triangles, quadrilaterals, tetrahedra, prisms, hexahedra) by automatically choosing the related element interpolation functions. For each process of a multi‐field problem, only a single instance of the finite element object is required. The element objects provide a flexible coding environment for multi‐field problems with different element types. Here, the C++ implementations of the objects are given and described in detail. The efficiency of the new element objects is demonstrated by several test cases dealing with thermo‐hydro‐mechanical (THM) coupled problems for geotechnical applications. Copyright © 2006 John Wiley & Sons, Ltd.     

New development in extended finite element modeling of large elasto‐plastic deformations

This paper presents new achievements in the extended finite element modeling of large elasto‐plastic deformation in solid problems. The computational technique is presented based on the extended finite element method (X‐FEM) coupled with the Lagrangian formulation in order to model arbitrary interfaces in large deformations. In X‐FEM, the material interfaces are represented independently of element boundaries, and the process is accomplished by partitioning the domain with some triangular sub‐elements whose Gauss points are used for integration of the domain of elements. The large elasto‐plastic deformation formulation is employed within the X‐FEM framework to simulate the non‐linear behavior of materials. The interface between two bodies is modeled by using the X‐FEM technique and applying the Heaviside‐ and level‐set‐based enrichment functions. Finally, several numerical examples are analyzed, including arbitrary material interfaces, to demonstrate the efficiency of the X‐FEM technique in large plasticity deformations. Copyright © 2008 John Wiley & Sons, Ltd.     

A continuum sensitivity method for finite thermo‐inelastic deformations with applications to the design of hot forming processes

A computational framework is presented to evaluate the shape as well as non‐shape (parameter) sensitivity of finite thermo‐inelastic deformations using the continuum sensitivity method (CSM). Weak sensitivity equations are developed for the large thermo‐mechanical deformation of hyperelastic thermo‐viscoplastic materials that are consistent with the kinematic, constitutive, contact and thermal analyses used in the solution of the direct deformation problem. The sensitivities are defined in a rigorous sense and the sensitivity analysis is performed in an infinite‐dimensional continuum framework. The effects of perturbation in the preform, die surface, or other process parameters are carefully considered in the CSM development for the computation of the die temperature sensitivity fields. The direct deformation and sensitivity deformation problems are solved using the finite element method. The results of the continuum sensitivity analysis are validated extensively by a comparison with those obtained by finite difference approximations (i.e. using the solution of a deformation problem with perturbed design variables). The effectiveness of the method is demonstrated with a number of applications in the design optimization of metal forming processes. Copyright © 2002 John Wiley & Sons, Ltd.     

An Eulerian finite‐volume scheme for large elastoplastic deformations in solids

Conservative formulations of the governing laws of elastoplastic solid media have distinct advantages when solved using high‐order shock capturing methods for simulating processes involving large deformations and shock waves. In this paper one such model is considered where inelastic deformations are accounted for via conservation laws for elastic strain with relaxation source terms. Plastic deformations are governed by the relaxation time of tangential stresses. Compared with alternative Eulerian conservative models, the governing system consists of fewer equations overall. A numerical scheme for the inhomogeneous system is proposed based upon the temporal splitting. In this way the reduced system of non‐linear elasticity is solved explicitly, with convective fluxes evaluated using high‐order approximations of Riemann problems locally throughout the computational mesh. Numerical stiffness of the relaxation terms at high strain rates is avoided by utilizing certain properties of the governing model and performing an implicit update. The methods are demonstrated using test cases involving large deformations and high strain rates in one‐, two‐, and three‐dimensions. Copyright © 2009 John Wiley & Sons, Ltd.     

FEM-BEM superposition method for fracture analysis of quasi-brittle structures

Fracture analysis of civil engineering structures often requires appropriate modeling of discrete cracks propagating in an inhomogeneous or nonlinear material. For example, quasi-brittle materials, such as concrete, are characterized by formation of cracks with fracture process zone under tension and plasticity under compression. Application of either finite element method (FEM) or boundary element method (BEM) to problems involving simultaneously discrete cracks and inhomogeneities or plastic deformations faces certain difficulties. Therefore, we propose the FEM-BEM superposition method, which removes the respective methods disadvantages while keeping their advantages. In the proposed method, the original problem involving both material inhomogeneity or plasticity and discrete cracks is decomposed into two subproblems. The inhomogeneity or inelastic deformation is represented in only one of the subproblems, while the cracks appear only in the other. The former subproblem is analyzed using FEM and the latter one by BEM, so as to utilize the advantages of the two methods. The solution of the original problem is then obtained by superposing the solutions of the two subproblems. In order to verify validity of the proposed method we present numerical results of several examples, including both linear-elastic and nonlinear fracture mechanics. The results are compared with available analytical solutions or with data computed by other numerical methods, showing both accuracy and computational superiority of the proposed method.     

Confidence structural robust optimization by non‐linear semidefinite programming‐based single‐level formulation

Structural robust optimization problems are often solved via the so‐called Bi‐level approach. This solution procedure often involves large computational efforts and sometimes its convergence properties are not so good because of the non‐smooth nature of the Bi‐level formulation. Another problem associated with the traditional Bi‐level approach is that the confidence of the robustness of the obtained solutions cannot be fully assured at least theoretically. In the present paper, confidence single‐level non‐linear semidefinite programming (NLSDP) formulations for structural robust optimization problems under stiffness uncertainties are proposed. This is achieved by using some tools such as S‐procedure and quadratic embedding for convex analysis. The resulted NLSDP problems are solved using the modified augmented Lagrange multiplier method which has sound mathematical properties. Numerical examples show that confidence robust optimal solutions can be obtained with the proposed approach effectively. Copyright © 2010 John Wiley & Sons, Ltd.     

Ductile damage analysis of elasto‐plastic shells at large inelastic strains

The objective of this contribution is to model ductile damage phenomena under consideration of large inelastic strains, to couple the corresponding constitutive law with a multi‐layer shell kinematics and to give finally an adequate finite element formulation. An elastic–plastic constitutive law is formulated by using a spatial hyperelasto‐plastic formulation based on the multiplicative decomposition of the deformation gradient. To include isotropic ductile damage the continuum damage model of Rousselier is modified so as to consider large strains and additionally extended by various void nucleation and macro‐crack criteria. In order to achieve numerical efficiency, elastic strains are supposed to be sufficiently small providing a numerical effective integration based on the backward Euler rule. Finite element formulation is enriched by means of the enhanced strain concept. Thus the well‐known deficiencies due to incompressible deformations and the inclusion of transverse strains are avoided. Several examples are given to demonstrate the performance of the algorithms developed concerning large inelastic strains of shells and ductile damage phenomena. Copyright © 2000 John Wiley & Sons, Ltd.     

A logarithmic‐exponential backward‐Euler‐based split of the flow rule for anisotropic inelastic behaviour at small elastic strain

A basic aspect of modern algorithmic formulations for large‐deformation hyperelastic‐based isotropic inelastic material models is the exponential backward‐Euler form of the algorithmic flow rule in the context of the multiplicative decomposition of the deformation gradient. Advantages of this approach in the isotropic context include the exact algorithmic fulfilment of inelastic incompressibility. The purpose of this short work is to show that such an algorithm can be formulated for anisotropic inelastic models as well under assumption of small elastic strain, i.e. for metals. In particular, the current approach works for both phenomenological anisotropy as well as for crystal plasticity. The major difference between the current and previous approaches lies in the fact that the elastic rotation is reduced algorithmically to a dependent internal variable, resulting in a smaller internal variable system. Copyright © 2006 John Wiley & Sons, Ltd.     

A finite‐volume method for deformation analysis of woven fabrics

Efficient numerical methods for simulating cloth deformations have been identified as the key to the development of successful Computer‐Aided Design systems for clothing products. This paper presents the formulation of a new finite‐volume method for the simulation of complex deformations of initially flat woven fabric sheets under self‐weight or externally applied loading. The fabric sheet is assumed to undergo very large displacements and rotations but small strains during the process of deformation. The fabric material is assumed to be linear elastic and orthotropic. The fabric sheet is discretized into many small structured patches called finite volumes (or control volumes), each containing one grid node and several face nodes. The bending and membrane deformations of a typical volume can be defined using the global co‐ordinates of its grid node and surrounding face nodes. The equilibrium equations governing the complex deformations are derived employing the principle of stationary total potential energy. These equations are solved using a single‐step full Newton–Raphson method which is found to be capable of predicting the final deformed shape, the only result of interest in a fabric drape analysis. To speed up convergence, the line search technique is adopted with good effect. This single‐step approach is more efficient than the step‐by‐step incremental approach employed in conventional non‐linear finite element analysis of load‐bearing structures. A number of example simulations of fabric drape/buckling deformations are included in the paper, which demonstrate the efficiency and validity of the proposed method. Copyright © 1999 John Wiley & Sons, Ltd.     

On semi‐analytical probabilistic finite element method for homogenization of the periodic fiber‐reinforced composites

The main aim of this paper is a development of the semi‐analytical probabilistic version of the finite element method (FEM) related to the homogenization problem. This approach is based on the global version of the response function method and symbolic integral calculation of basic probabilistic moments of the homogenized tensor and is applied in conjunction with the effective modules method. It originates from the generalized stochastic perturbation‐based FEM, where Taylor expansion with random parameters is not necessary now and is simply replaced with the integration of the response functions. The hybrid computational implementation of the system MAPLE with homogenization‐oriented FEM code MCCEFF is invented to provide probabilistic analysis of the homogenized elasticity tensor for the periodic fiber‐reinforced composites. Although numerical illustration deals with a homogenization of a composite with material properties defined as Gaussian random variables, other composite parameters as well as other probabilistic distributions may be taken into account. The methodology is independent of the boundary value problem considered and may be useful for general numerical solutions using finite or boundary elements, finite differences or volumes as well as for meshless numerical strategies. Copyright © 2011 John Wiley & Sons, Ltd.     

Topology optimization for nano‐scale heat transfer

We consider the problem of optimal design of nano‐scale heat conducting systems using topology optimization techniques. At such small scales the empirical Fourier's law of heat conduction no longer captures the underlying physical phenomena because the mean‐free path of the heat carriers, phonons in our case, becomes comparable with, or even larger than, the feature sizes of considered material distributions. A more accurate model at nano‐scales is given by kinetic theory, which provides a compromise between the inaccurate Fourier's law and precise, but too computationally expensive, atomistic simulations. We analyze the resulting optimal control problem in a continuous setting, briefly describing the computational approach to the problem based on discontinuous Galerkin methods, algebraic multigrid preconditioned generalized minimal residual method, and a gradient‐based mathematical programming algorithm. Numerical experiments with our implementation of the proposed numerical scheme are reported. Copyright © 2008 John Wiley & Sons, Ltd.     

A framework for implementation of RVE‐based multiscale models in computational homogenization using isogeometric analysis

This study presents an isogeometric framework for incorporating representative volume element–based multiscale models into computational homogenization. First‐order finite deformation homogenization theory is derived within the framework of the method of multiscale virtual power, and Lagrange multipliers are used to illustrate the effects of considering different kinematical constraints. Using a Lagrange multiplier approach in the numerical implementation of the discrete system naturally leads to a consolidated treatment of the commonly employed representative volume element boundary conditions. Implementation of finite deformation computational strain‐driven, stress‐driven, and mixed homogenization is detailed in the context of isogeometric analysis (IGA), and performance is compared to standard finite element analysis. As finite deformations are considered, a numerical multiscale stability analysis procedure is also detailed for use with IGA. Unique implementation aspects that arise when computational homogenization is performed using IGA are discussed, and the developed framework is applied to a complex curved microstructure representing an architectured material.     

Finite element methods for the non‐linear mechanics of crystalline sheets and nanotubes

The formulation and finite element implementation of a finite deformation continuum theory for the mechanics of crystalline sheets is described. This theory generalizes standard crystal elasticity to curved monolayer lattices by means of the exponential Cauchy–Born rule. The constitutive model for a two‐dimensional continuum deforming in three dimensions (a surface) is written explicitly in terms of the underlying atomistic model. The resulting hyper‐elastic potential depends on the stretch and the curvature of the surface, as well as on internal elastic variables describing the rearrangements of the crystal within the unit cell. Coarse grained calculations of carbon nanotubes (CNTs) are performed by discretizing this continuum mechanics theory by finite elements. A smooth discrete representation of the surface is required, and subdivision finite elements, proposed for thin‐shell analysis, are used. A detailed set of numerical experiments, in which the continuum/finite element solutions are compared to the corresponding full atomistic calculations of CNTs, involving very large deformations and geometric instabilities, demonstrates the accuracy of the proposed approach. Simulations for large multi‐million systems illustrate the computational savings which can be achieved. Copyright © 2003 John Wiley & Sons, Ltd.     

A co‐rotational grid‐based model for fabric drapes

Fabric drapes are typical large displacement, large rotation and small strain problems. Compared to conventional geometric non‐linear shell analyses, computational fabric drape analysis is particularly challenging due to the extremely weak bending rigidities of fabrics. Compared to continuum shell finite element methods, grid‐ or particle‐based methods appear to be more successful in high drapeability problems. The latter methods often resort to simple particle mechanics and formulate the elastic energy in terms of the inter‐particle distances and trigonometrical functions of the angles between the straight lines joining adjacent particles. In this paper, the co‐rotational approach and commonly employed assumptions for small strain problems in finite element analysis will be adopted to formulate the elastic energy. It will be seen that the internal force vector and the stiffness matrix are considerably simpler than other grid‐based models, yet the sparsity of the tangential stiffness matrix remains unchanged. A number of examples are considered and the predicted results are promising. Copyright © 2003 John Wiley & Sons, Ltd.     

Modelling of reinforced‐concrete structures providing crack‐spacing based on X‐FEM,ED‐FEM and novel operator split solution procedure

In this work, we present a novel approach to the finite element modelling of reinforced‐concrete (RC) structures that provides the details of the constitutive behavior of each constituent (concrete, steel and bond‐slip), while keeping formally the same appearance as the classical finite element model. Each component constitutive behavior can be brought to fully non‐linear range, where we can consider cracking (or localized failure) of concrete, the plastic yielding and failure of steel bars and bond‐slip at concrete steel interface accounting for confining pressure effects. The standard finite element code architecture is preserved by using embedded discontinuity (ED‐FEM) and extended (X‐FEM) finite element strain representation for concrete and slip, respectively, along with the operator split solution method that separates the problem into computing the deformations of RC (with frozen slip) and the current value of the bond‐slip. Several numerical examples are presented in order to illustrate very satisfying performance of the proposed methodology. Copyright © 2010 John Wiley & Sons, Ltd.     

Numerical p‐version refinement studies for the regularized stress‐BEM

In the development of the boundary element method (BEM) and the finite element method (FEM) researchers have typically selected similar basis functions. That is, both methods typically employ low‐order interpolations such as piece‐wise linear or piece‐wise quadratic and rely on h‐version refinement to increase accuracy as required. In the case of the FEM, the decision to use low‐order elements is made for computational efficiency as an attractive compromise between local modeling accuracy and sparseness of the resulting linear system. However, in many BEM formulations, low‐order elements may be the only practical choice given the complexity of using analytic integration formulae in conjunction with special integral interpretations. Unlike their efficient use in the FEM, fine meshes of low‐order elements in the BEM are highly inefficient from a computational standpoint given the dense nature of BEM systems. Moreover, owing to singularities in the kernel functions, the BEM should be expected to benefit more so than the FEM from very high levels of local accuracy. Through the use of regularized algorithms which only require numerical integration, p‐version refinement in the BEM is easily extended to include any set of basis functions with no significant increase in programming complexity. Numerical results show that by using interpolations as high as 12th and 16th order, one can expect reductions in error by as many as five orders of magnitude over comparable algorithms based on similar system size. For two‐dimensional problems, it is also shown that, for a given level of error, one can expect reductions in system size by an order of magnitude, thus leading to a reduction in computational expense for conventional algorithms by three orders of magnitude. Copyright © 2003 John Wiley & Sons, Ltd.     

A local multigrid X‐FEM strategy for 3‐D crack propagation

A multiscale method for 3‐D crack propagation simulation in large structures is proposed. The method is based on the extended finite element method (X‐FEM). The asymptotic behavior of the crack front is accurately modeled using enriched elements and no remeshing is required during crack propagation. However, the different scales involved in fracture mechanics problems can differ by several orders of magnitude and industrial meshes are usually not designed to account for small cracks. Enrichments are therefore useless if the crack is too small compared with the element size. To overcome this drawback, a project combining different numerical techniques was started. The first step was the implementation of a global multigrid algorithm within the X‐FEM framework and was presented in a previous paper (Eur. J. Comput. Mech. 2007; 16 :161–182). This work emphasized the high efficiency in cpu time but highlighted that mesh refinement is required on localized areas only (cracks, inclusions, steep gradient zones). This paper aims at linking the different scales by using a local multigrid approach. The coupling of this technique with the X‐FEM is described and computational aspects dealing with intergrid operators, optimal multiscale enrichment strategy and level sets are pointed out. Examples illustrating the accuracy and efficiency of the method are given. Copyright © 2008 John Wiley & Sons, Ltd.     

Numerical homogenization of N‐component composites including stochastic interface defects

The purpose of this paper is to present a mathematical formulation and numerical analysis for a homogenization problem of random elastic composites with stochastic interface defects. The homogenization of composites so defined is carried out in two steps: (i) probabilistic averaging of stochastic discontinuities in the interphase region, (ii) probabilistic homogenization by extending the effective modules method to media random in the micro‐scale. To obtain such an approach the classical mathematical homogenization method is formulated for n‐component composite with random elastic components and implemented in the FEM‐based computer program. The article contains also numerous computational experiments illustrating stochastic sensitivity of the model to interface defects parameters and verifying statistical convergence of probabilistic simulation procedure. Copyright © 2000 John Wiley & Sons, Ltd.     

An X‐FEM approach for large sliding contact along discontinuities

The extended finite element method (X‐FEM) has been developed to minimize requirements on the mesh design in a problem with a displacement discontinuity. This advantage, however, still remains limited to the small deformation hypothesis when considering sliding discontinuities. The approach presented in this paper proposes to couple X‐FEM with a Lagrangian large sliding frictionless contact algorithm. A new hybrid X‐FEM contact element was developed with a contact search algorithm allowing for an update of contacting surfaces pairing. The stability of the contact formulation is ensured by an algorithm for fulfilling Ladyzhenskaya‐Babuska‐Brezzi (LBB) condition. Several 2D simple examples are presented in this paper in order to prove its efficiency and stability. Copyright © 2008 John Wiley & Sons, Ltd.