International Journal for Numerical Methods in Engineering
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Dimensional reduction of nonlinear finite element dynamic models with finite rotations and energy‐based mesh sampling and weighting for computational efficiency SUMMARY A rigorous computational framework for the dimensional reduction of discrete, high‐fidelity, nonlinear, finite element structural dynamics models is presented. It is based on the pre‐computation of solution snapshots, their compression into a reduced‐order basis, and the Galerkin projection of the given discrete high‐dimensional model onto this basis. To this effect, this framework distinguishes between vector‐valued displacements and manifold‐valued finite rotations. To minimize computational complexity, it also differentiates between the cases of constant and configuration‐dependent mass matrices. Like most projection‐based nonlinear model reduction methods, however, its computational efficiency hinges not only on the ability of the constructed reduced‐order basis to capture the dominant features of the solution of interest but also on the ability of this framework to compute fast and accurate approximations of the projection onto a subspace of tangent matrices and/or force vectors. The computation of the latter approximations is often referred to in the literature as hyper reduction. Hence, this paper also presents the energy‐conserving sampling and weighting (ECSW) hyper reduction method for discrete (or semi‐discrete), nonlinear, finite element structural dynamics models. Based on mesh sampling and the principle of virtual work, ECSW is natural for finite element computations and preserves an important energetic aspect of the high‐dimensional finite element model to be reduced. Equipped with this hyper reduction procedure, the aforementioned Galerkin projection framework is first demonstrated for several academic but challenging problems. Then, its potential for the effective solution of real problems is highlighted with the realistic simulation of the transient response of a vehicle to an underbody blast event. For this problem, the proposed nonlinear model reduction framework reduces the CPU time required by a typical high‐dimensional model by up to four orders of magnitude while maintaining a good level of accuracy. Copyright © 2014 John Wiley & Sons, Ltd.
International Journal for Numerical Methods in Engineering - Tập 98 Số 9 - Trang 625-662 - 2014
A higher order compact finite difference algorithm for solving the incompressible Navier–Stokes equations Abstract On the basis of the projection method, a higher order compact finite difference algorithm, which possesses a good spatial behavior, is developed for solving the 2D unsteady incompressible Navier–Stokes equations in primitive variable. The present method is established on a staggered grid system and is at least third‐order accurate in space. A third‐order accurate upwind compact difference approximation is used to discretize the non‐linear convective terms, a fourth‐order symmetrical compact difference approximation is used to discretize the viscous terms, and a fourth‐order compact difference approximation on a cell‐centered mesh is used to discretize the first derivatives in the continuity equation. The pressure Poisson equation is approximated using a fourth‐order compact difference scheme constructed currently on the nine‐point 2D stencil. New fourth‐order compact difference schemes for explicit computing of the pressure gradient are also developed on the nine‐point 2D stencil. For the assessment of the effectiveness and accuracy of the method, particularly its spatial behavior, a problem with analytical solution and another one with a steep gradient are numerically solved. Finally, steady and unsteady solutions for the lid‐driven cavity flow are also used to assess the efficiency of this algorithm. Copyright © 2011 John Wiley & Sons, Ltd.
International Journal for Numerical Methods in Engineering - Tập 88 Số 6 - Trang 511-532 - 2011
A phase‐field model for cohesive fracture SUMMARY In this paper, a phase‐field model for cohesive fracture is developed. After casting the cohesive zone approach in an energetic framework, which is suitable for incorporation in phase‐field approaches, the phase‐field approach to brittle fracture is recapitulated. The approximation to the Dirac function is discussed with particular emphasis on the Dirichlet boundary conditions that arise in the phase‐field approximation. The accuracy of the discretisation of the phase field, including the sensitivity to the parameter that balances the field and the boundary contributions, is assessed at the hand of a simple example. The relation to gradient‐enhanced damage models is highlighted, and some comments on the similarities and the differences between phase‐field approaches to fracture and gradient‐damage models are made. A phase‐field representation for cohesive fracture is elaborated, starting from the aforementioned energetic framework. The strong as well as the weak formats are presented, the latter being the starting point for the ensuing finite element discretisation, which involves three fields: the displacement field, an auxiliary field that represents the jump in the displacement across the crack, and the phase field. Compared to phase‐field approaches for brittle fracture, the modelling of the jump of the displacement across the crack is a complication, and the current work provides evidence that an additional constraint has to be provided in the sense that the auxiliary field must be constant in the direction orthogonal to the crack. The sensitivity of the results with respect to the numerical parameter needed to enforce this constraint is investigated, as well as how the results depend on the orders of the discretisation of the three fields. Finally, examples are given that demonstrate grid insensitivity for adhesive and for cohesive failure, the latter example being somewhat limited because only straight crack propagation is considered. Copyright © 2013 John Wiley & Sons, Ltd.
International Journal for Numerical Methods in Engineering - Tập 96 Số 1 - Trang 43-62 - 2013
An augmented cohesive zone element for arbitrary crack coalescence and bifurcation in heterogeneous materials Abstract We demonstrate that traditional cohesive zone (CZ) elements cannot be accurate when used in conjunction with solid elements with arbitrary intra‐element cracking capability, because they cannot capture the load transfer between cohesive interfaces and the solid elements when crack bifurcation or coalescence occurs. An augmented cohesive zone (ACZ) element based on the augmented finite element method formulation is therefore proposed. The new element allows for arbitrary separation of the cohesive element in accordance with the crack configuration of the abutting solid elements, thus correctly maintaining the non‐linear coupling between merging or bifurcating cracks. Numerical accuracy and effectiveness of the proposed ACZ element are demonstrated through several examples. Copyright © 2011 John Wiley & Sons, Ltd.
International Journal for Numerical Methods in Engineering - Tập 88 Số 9 - Trang 841-861 - 2011
Finite element interface models for the delamination analysis of laminated composites: mechanical and computational issues Abstract The finite element analysis of delamination in laminated composites is addressed using interface elements and an interface damage law. The principles of linear elastic fracture mechanics are indirectly used by equating, in the case of single‐mode delamination, the area underneath the traction/relative displacement curve to the critical energy release rate of the mode under examination. For mixed‐mode delamination an interaction model is used which can fulfil various fracture criteria proposed in the literature. It is then shown that the model can be recast in the framework of a more general damage mechanics theory. Numerical results are presented for the analyses of a double cantilever beam specimen and for a problem involving multiple delamination for which comparisons are made with experimental results. Issues related with the numerical solution of the non‐linear problem of the delamination are discussed, such as the influence of the interface strength on the convergence properties and the final results, the optimal choice of the iterative matrix in the predictor and the number of integration points in the interface elements. Copyright © 2001 John Wiley & Sons, Ltd.
International Journal for Numerical Methods in Engineering - Tập 50 Số 7 - Trang 1701-1736 - 2001
Linear and nonlinear solvers for variational phase-field models of brittle fracture
International Journal for Numerical Methods in Engineering - Tập 109 Số 5 - Trang 648-667 - 2017
Phase‐field modeling and simulation of fracture in brittle materials with strongly anisotropic surface energy Summary Crack propagation in brittle materials with anisotropic surface energy is important in applications involving single crystals, extruded polymers, or geological and organic materials. Furthermore, when this anisotropy is strong, the phenomenology of crack propagation becomes very rich, with forbidden crack propagation directions or complex sawtooth crack patterns. This problem interrogates fundamental issues in fracture mechanics, including the principles behind the selection of crack direction. Here, we propose a variational phase‐field model for strongly anisotropic fracture, which resorts to the extended Cahn‐Hilliard framework proposed in the context of crystal growth. Previous phase‐field models for anisotropic fracture were formulated in a framework only allowing for weak anisotropy. We implement numerically our higher‐order phase‐field model with smooth local maximum entropy approximants in a direct Galerkin method. The numerical results exhibit all the features of strongly anisotropic fracture and reproduce strikingly well recent experimental observations. Copyright © 2014 John Wiley & Sons, Ltd.
International Journal for Numerical Methods in Engineering - Tập 102 Số 3-4 - Trang 711-727 - 2015
Thermodynamically consistent phase‐field models of fracture: Variational principles and multi‐field FE implementations Abstract The computational modeling of failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in situations with complex crack topologies. This can be overcome by a diffusive crack modeling based on the introduction of a crack phase‐field. In this paper, we outline a thermodynamically consistent framework for phase‐field models of crack propagation in elastic solids, develop incremental variational principles and consider their numerical implementations by multi‐field finite element methods. We start our investigation with an intuitive and descriptive derivation of a regularized crack surface functional that Γ‐converges for vanishing length‐scale parameter to a sharp crack topology functional. This functional provides the basis for the definition of suitable convex dissipation functions that govern the evolution of the crack phase‐field. Here, we propose alternative rate‐independent and viscous over‐force models that ensure the local growth of the phase‐field. Next, we define an energy storage function whose positive tensile part degrades with increasing phase‐field. With these constitutive functionals at hand, we derive the coupled balances of quasi‐static stress equilibrium and gradient‐type phase‐field evolution in the solid from the argument of virtual power. Here, we consider a canonical two‐field setting for rate‐independent response and a time‐regularized three‐field formulation with viscous over‐force response. It is then shown that these balances follow as the Euler equations of incremental variational principles that govern the multi‐field problems. These principles make the proposed formulation extremely compact and provide a perfect base for the finite element implementation, including features such as the symmetry of the monolithic tangent matrices. We demonstrate the performance of the proposed phase‐field formulations of fracture by means of representative numerical examples. Copyright © 2010 John Wiley & Sons, Ltd.
International Journal for Numerical Methods in Engineering - Tập 83 Số 10 - Trang 1273-1311 - 2010
Calculation of eigenpair derivatives for asymmetric damped systems with distinct and repeated eigenvalues
International Journal for Numerical Methods in Engineering - Tập 103 Số 7 - Trang 501-515 - 2015
Eigensolution reanalysis of modified structures using epsilon‐algorithm Abstract Based on the Neumann series expansion and epsilon‐algorithm, a new eigensolution reanalysis method is developed. In the solution process, the basis vectors can be obtained using the matrix perturbation or the Neumann series expansion to construct the vector sequence, and then using the epsilon algorithm table to obtain the approximate eigenvectors. The approximate eigenvalues are computed from the Rayleigh quotients. The solution steps are straightforward and it is easy to implement with the general finite element analysis system. Two numerical examples, a 40‐storey frame and a chassis structure, are given to demonstrate the application of the present method. By comparing with the exact solutions and the Kirsch method solutions, it is shown that the excellent results are obtained for very large changes in the design, and that the accuracy of the epsilon‐algorithm is higher than that of the Kirsch method and the computation time is less than that of the Kirsch method. Copyright © 2005 John Wiley & Sons, Ltd.
International Journal for Numerical Methods in Engineering - Tập 66 Số 13 - Trang 2115-2130 - 2006
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