Strict upper error bounds on computed outputs of interest in computational structural mechanics

This paper introduces new strict error bounds of computed outputs of interest for time-dependent nonlinear problems in quasi-statics as well as in dynamics. All sources of errors, including modeling errors, are taken into account. Therefore, such error bounds are also suitable tools for analyzing various approximations, particularly in dynamics. Small-displacement problems without softening, such as (visco)plasticity problems, are included through the classical thermodynamics framework involving internal state variables; the material models are not necessarily standard.     

On the duality of finite element discretization error control in computational Newtonian and Eshelbian mechanics

In this paper, goal-oriented a posteriori error estimators of the averaging type are presented for the error obtained while approximately evaluating theJ-integral in nonlinear elastic fracture mechanics. Since the value of the J-integral is one component of the material force acting on the crack tip of a pre-cracked elastic body, the appropriate mechanical framework to be chosen is the one named after Eshelby rather than classical Newtonian mechanics. However, in a finite element setting, the discretized Eshelby problem is generally not solved explicitly. Rather, its solution is approximated by the finite element solution of the corresponding discretized dual Newton problem. As a consequence, discrete material forces arise not only at the crack tip but also at other nodes of the current finite element mesh. It is the objective of this paper to establish goal-oriented a posteriori error estimators in both the framework of Eshelbian and Newtonian mechanics and to elaborate their dual relations. This allows to control the error of the J-integral while, at the same time, no further discrete material forces arise during the adaptive mesh refinement process which could lead to misleading mechanical interpretations of the results obtained by the finite element method. The paper is concluded by numerical examples that illustrate our theoretical results. Dedicated to the memory of the esteemed colleague Professor Karl Popp, University of Hannover, who unexpectedly passed away on April 24, 2005.     

Advances in computational contact mechanics

Great strides have recently been made in the application of computational mechanics to the design of highly complex engineering systems. It has now become abundantly clear that advanced modelling techniques are central to the competitiveness of the industrialised nations. Excellent examples of this assertion are the computer-integrated design of the recent Boeing 777 aircraft, the collapsible foam-filled structures for the car of the next century and new prosthetic implants for Rheumatoid Arthritis. It is with this in mind that the author focuses his attention to a class of problems where contact mechanics plays a major role in dictating the mechanical integrity of the component/system. Three aspects of the current study are accordingly examined. The first is concerned with the development of the appropriate dynamic variational inequalities expressions, which are capable of the accurate and consistent representation of contact problems. The second is concerned with the development of robust solution algorithms that guarantee the accurate imposition of the kinematic contact constraint and avoid interpenetration. The third is concerned with the application of the developed algorithms to realistic design problems involving intricate mechanical and biomechanical systems.     

Some computational and algorithmic developments in computational mechanics of discontinua

Discontinua simulations are becoming an important part of computational mechanics to the extent that computational mechanics of discontinua is becoming a separate sub-discipline of computational mechanics. Among the most widely used methods of computational mechanics of discontinua are discrete-element methods, combined finite-discrete-element methods and discontinuum deformation analysis methods. A range of key algorithmic procedures is common to most of these methods. These include contact detection, explicit solvers, fracture and fragmentation models, handling of complex geometric considerations when processing interaction in three dimensions (contact kinematics) and fluid coupling. In recent years, there have been major breakthroughs in almost all of these key algorithmic aspects. These include linear contact-detection procedures (NBS, C-grid), discretized contact solutions, fracture and fragmentation solutions, together with fluid pressure driven fracture process and three-dimensional explicit solvers incorporating finite rotations. Many of these breakthroughs have not yet been applied across the full range of relevant discontinuum problems. The major reason for this is that discrete-element method, discontinuum deformation analysis and combined finite-discrete-element method publications are spread over a wide range of specialist journals and conferences. Thus in this paper, the main features of a selection of algorithmic breakthroughs are reviewed for the first time, enabling researchers in different fields to apply these compatible developments to their specific applications.     

Material state remapping in computational solid mechanics

A computational procedure for remapping material state information from one finite element mesh to another is described. The procedure is useful in connection with evolving meshes for inelastic problems, as for example occur in the context of fracture simulation and adaptive mesh refinement. The proposed method is based on weak enforcement of equality between corresponding fields on the two meshes, where piecewise‐constant fields on both meshes are generalized from the quadrature‐point values. The essential algorithmic problem is that of calculating the volume partition of an arbitrary convex region with respect to a covering set of disjoint convex regions. Instead of geometrically resolving the associated intersections, the problem is herein approximated by a constrained optimization problem, which may be readily and efficiently solved computationally. This formulation is a main contribution of the paper. Computational examples are given that illustrate the effectiveness of the proposed procedure. Copyright © 2002 John Wiley & Sons, Ltd.     

Isoparametric finite point method in computational mechanics

In this paper, a new meshless method, the isoparametric finite point method (IFPM) in computational mechanics is presented. The present IFPM is a truly meshless method and developed based on the concepts of meshless discretization and local isoparametric interpolation. In IFPM, the unknown functions, their derivatives, and the sub-domain and its boundaries of an arbitrary point are described by the same shape functions. Two kinds of shape functions that satisfy the Kronecker-Delta property are developed for the scattered points in the domain and on the boundaries, respectively. Conventional point collocation method is employed for the discretization of the governing equation and the boundary conditions. The essential (Dirichlet) and natural (Neumann) boundary conditions can be directly enforced at the boundary points. Several numerical examples are presented together with the results obtained by the exact solution and the finite element method. The numerical results show that the present IFPM is a simple and efficient method in computational mechanics.     

A Trefftz approach to computational mechanics

This paper presents a Trefftz method for solving structural elasticity problems and flow problems of incompressible viscous fluids. The problem of unilateral contact is also dealt with. For each type of problem, Trefftz polynomials and associated variational formulations are given. Complex structures are studied by a sub‐structuring technique. This method requires the resolution of a non‐symmetrical linear system. It is shown that it is possible to take advantage of this Trefftz approximation in two ways: (i) the approach presented can be considered as a simplified method which enables a solution to be evaluated quickly; (ii) this approach also makes it possible to obtain a good quality solution associated with high degree polynomial bases. This method is adapted to optimization processes because the discretization of the structure requires only very few sub‐domains to build a good approximation and offers a great flexibility in use. Copyright © 2003 John Wiley & Sons, Ltd.     

Goal-oriented explicit residual-type error estimates in XFEM

A goal-oriented a posteriori error estimator is derived to control the error obtained while approximately evaluating a quantity of engineering interest, represented in terms of a given linear or nonlinear functional, using extended finite elements of $Q1$ type. The same approximation method is used to solve the dual problem as required for the a posteriori error analysis. It is shown that for both problems to be solved numerically the same singular enrichment functions can be used. The goal-oriented error estimator presented can be classified as explicit residual type, i.e. the residuals of the approximations are used directly to compute upper bounds on the error of the quantity of interest. This approach therefore extends the explicit residual-type error estimator for classical energy norm error control as recently presented in Gerasimov et al. (Int J Numer Meth Eng 90:1118–1155, 2012a). Without loss of generality, the a posteriori error estimator is applied to the model problem of linear elastic fracture mechanics. Thus, emphasis is placed on the fracture criterion, here the $J$ -integral, as the chosen quantity of interest. Finally, various illustrative numerical examples are presented where, on the one hand, the error estimator is compared to its finite element counterpart and, on the other hand, improved enrichment functions, as introduced in Gerasimov et al. (2012b), are discussed.     

Knowledge-based methods and smart algorithms in computational mechanics

Effective methods leading to automated, computer-based solution of complex engineering design problems are studied in this paper. In particular, methods of automation of the finite element analyses are of primary interest here. These include algorithmic approaches, based on error estimation, adaptivity and smart algorithms, as well as heuristic approaches based on methods of knowledge engineering. A computational environment, which interactively couples h-p adaptive finite element methods with object-oriented programming and expert system tools, is presented. Several examples illustrate the merit and potential of the approaches studied here.     

Validation and error estimation of computational models

This paper develops a Bayesian methodology for assessing the confidence in model prediction by comparing the model output with experimental data when both are stochastic. The prior distribution of the response is first computed, which is then updated based on experimental observation using Bayesian analysis to compute a validation metric. A model error estimation methodology is then developed to include model form error, discretization error, stochastic analysis error (UQ error), input data error and output measurement error. Sensitivity of the validation metric to various error components and model parameters is discussed. A numerical example is presented to illustrate the proposed methodology.     

Assessing the error in bootstrap estimates with dependent data

Bootstrap estimates, like most random variables, are subject to sampling variation. Efron and Tibshirani (1993) studied the variability in bootstrap estimates with independent data. Efron (1992) proposed the jackknife-after-bootstrap, a method for estimating the variability from the bootstrap samples themselves. We address the issue of studying the variability in bootstrap estimates for dependent data. We modify Efron's method to render it suitable to operate through the block bootstrap. A simulation study is carried out to investigate the consistency of the modified method. The performance of this method is judged by using the same setting as that used by Efron and Tibshirani (1993). Our results confirm that this method is reliable and has an advantage in the context of dependent data.     

Integral estimates to the fast brittle fracture mechanics

In the terms of a general approach the problem of the fast crack propagation in elastic bodies is considered. The time dependence of, so called domain of possible fractures due to the fast crack propagation, is obtained. The dynamic domain of possible fractures is the upper estimate of the zone of real fractures and thereby it takes into account such important features of the fast brittle fracture as the instability of the fast crack propagation, branching, waving, the influence of boundary conditions and so on. The problem of dynamic crack initiation is considered in some details. Within the linear fracture mechanics the dynamic criterion of crack initiation is offered.     

Adaptive mesh refinement of the boundary element method for potential problems by using mesh sensitivities as error indicators

This paper presents a novel method for error estimation and h-version adaptive mesh refinement for potential problems which are solved by the boundary element method (BEM). Special sensitivities, denoted as mesh sensitivities, are used to evaluate a posteriori error indicators for each element, and a global error estimator. A mesh sensitivity is the sensitivity of a physical quantity at a boundary node with respect to perturbation of the mesh. The element error indicators for all the elements can be evaluated from these mesh sensitivities. Mesh refinement can then be performed by using these element error indicators as guides.The method presented here is suitable for both potential and elastostatics problems, and can be applied for adaptive mesh refinement with either linear or quadratic boundary elements. For potential problems, the physical quantities are potential and/or flux; for elastostatics problems, the physical quantities are tractions/displacements (or tangential derivatives of displacements). In this paper, the focus is on potential problems with linear elements, and the proposed method is validated with two illustrative examples. However, it is easy to extend these ideas to elastostatics problems and to quadratic elements.The computing for this research has been supported by the Cornell National Supercomputer Facility.     

A-posteriori error estimates for the finite element method

Computable a-posteriori error estimates for finite element solutions are derived in an asymptotic form for h → 0 where h measures the size of the elements. The approach has similarity to the residual method but differs from it in the use of norms of negative Sobolev spaces corresponding to the given bilinear (energy) form. For clarity the presentation is restricted to one-dimensional model problems. More specifically, the source, eigenvalue, and parabolic problems are considered involving a linear, self-adjoint operator of the second order. Generalizations to more general one-dimensional problems are straightforward, and the results also extend to higher space dimensions; but this involves some additional considerations. The estimates can be used for a practical a-posteriori assessment of the accuracy of a computed finite element solution, and they provide a basis for the design of adaptive finite element solvers.     

Strong discontinuities in antiplane/torsional problems of computational failure mechanics

This paper considers the analysis of localized failures and fracture of solids under antiplane conditions. We consider the longitudinal cracking of shafts in torsion, with the crack propagating through the cross section, besides pure antiplane problems (that is, with loading perpendicular to the plane of analysis). The main goal is the theoretical characterization and the numerical resolution of strong discontinuities in this setting, that is, discontinuities of the antiplane displacement field modeling the cracks. A multi-scale framework is considered, by which the discontinuities are treated locally in the (global) antiplane mechanical boundary value problem of interest, incorporating effectively the contributions of the discontinuities to the failure of the solid. We can identify among these contributions, besides the change in the stiffness of the solid or structural member, the localized energy dissipation associated to the cohesive law governing the physical response of the discontinuity surfaces. A main outcome of this approach is the development of new finite elements with embedded discontinuities for the antiplane problem that capture these solutions, and physical effects, locally at the element level. This local structure allows the static condensation at the element level of the degrees of freedom considered in the approximation of the antiplane displacement jumps along the discontinuity. In this way, the new elements result not only in a cost efficient computational tool of analysis for these problems, but also in a technique that can be easily incorporated in an existing finite element code, while resolving objectively those physical dissipative effects along the localized surfaces of failure. We develop, in particular, quadrilateral finite elements with the embedded discontinuities exhibiting constant and linear approximations of the displacement jumps, showing the superior performance of the latter given the stress locking associated with quadrilateral elements with constant jumps only. This limitation manifests itself in spurious transfers of stresses across the discontinuities, leading to severe oscillations in the stress field and an overall excessively stiff solution of the problem. These features are illustrated with several numerical examples, including convergence tests and validations with analytical results existing in the literature, showing in the process the treatment of characteristic situations like snap-backs, commonly encountered in the modeling of these structural members at failure.     

Recovering lower bounds of the error by postprocessing implicit residual a posteriori error estimates

Classical residual type error estimators approximate the error flux around the elements and yield upper bounds of the exact (or reference) error. Lower bounds of the error are also needed in goal oriented adaptivity and for bounds on functional outputs. This work introduces a simple and cheap strategy to recover a lower bound estimate from standard upper bound estimates. This lower bound may also be used to assess the effectivity of the former estimate and to improve it. Copyright © 2003 John Wiley & Sons, Ltd.     

Equilibrated patch recovery error estimates: simple and accurate upper bounds of the error

This paper introduces a new recovery‐type error estimator ensuring local equilibrium and yielding a guaranteed upper bound of the error. The upper bound property requires the recovered solution to be both statically equilibrated and continuous. The equilibrium is obtained locally (patch‐by‐patch) and the continuity is enforced by a postprocessing based on the partition of the unity concept. This postprocess is expected to preserve the features of the locally equilibrated stress field. Nevertheless, the postprocess phase modifies the equilibrium, which is no longer exactly fulfilled. A new methodology is introduced that yields upper bound estimates by taking into account this lack of equilibrium. This requires computing the ??₂ norm of the error or relating it with the energy norm. The guaranteed upper bounds are obtained by using a pessimistic bound of the error ??₂ norm, derived from an eigenvalue problem. Nevertheless, these bounds are not sharp. An additional strategy based on a more accurate assessment of the error ??₂ norm is introduced, providing sharp estimates, which are practical upper bounds as it is demonstrated in the numerical tests. Copyright © 2006 John Wiley & Sons, Ltd.