A stable discontinuity-enriched finite element method for 3-D problems containing weak and strong discontinuities

Zhang, Jian; Boom, Sanne J. van den; Keulen, Fred van; Aragón, Alejandro M.

doi:10.1016/j.cma.2019.05.018

Cited by 28 publications

(33 citation statements)

References 45 publications

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“…The use of enriched finite element methods in topology optimization approaches is not new; the eXtended/Generalized Finite Element Method (X/GFEM) (Oden et al 1998;Moës et al 1999;Moës et al 2003;Belytschko et al 2009;Aragón et al 2010), for example, has been explored in this context. However, the Interface-enriched Generalized Finite Element Method (IGFEM) has been shown to have many advantages over X/GFEM (Soghrati et al 2012a;van den Boom et al 2019a). In this work, we extend IGFEM to be used in a level set-based topology optimization framework.…”

Section: Introductionmentioning

confidence: 99%

“…Furthermore, enrichment functions are exactly zero at original mesh nodes. Therefore, original mesh nodes retain their physical meaning and essential boundary conditions can be enforced directly on non-matching edges (Cuba-Ramos et al 2015;Aragón and Simone 2017;van den Boom et al 2019a). It was shown that IGFEM is optimally convergent under mesh refinement for problems without singularities (Soghrati et al 2012a, b).…”

Section: Introductionmentioning

confidence: 99%

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An interface-enriched generalized finite element method for level set-based topology optimization

Boom

Zhang

Keulen

et al. 2020

Struct Multidisc Optim

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During design optimization, a smooth description of the geometry is important, especially for problems that are sensitive to the way interfaces are resolved, e.g., wave propagation or fluid-structure interaction. A level set description of the boundary, when combined with an enriched finite element formulation, offers a smoother description of the design than traditional density-based methods. However, existing enriched methods have drawbacks, including ill-conditioning and difficulties in prescribing essential boundary conditions. In this work, we introduce a new enriched topology optimization methodology that overcomes the aforementioned drawbacks; boundaries are resolved accurately by means of the Interface-enriched Generalized Finite Element Method (IGFEM), coupled to a level set function constructed by radial basis functions. The enriched method used in this new approach to topology optimization has the same level of accuracy in the analysis as the standard finite element method with matching meshes, but without the need for remeshing. We derive the analytical sensitivities and we discuss the behavior of the optimization process in detail. We establish that IGFEM-based level set topology optimization generates correct topologies for well-known compliance minimization problems.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

An interface-enriched generalized finite element method for level set-based topology optimization

Boom

Zhang

Keulen

et al. 2020

Struct Multidisc Optim

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“…Enriched nodes need to be placed along the boundary, and integration elements need to be created. These geometric operations can be done efficiently but require specialized code: a geometric engine The hierarchical construction of enrichment functions calls for a dedicated data structure, such as an ordered tree, to store the hierarchy.…”

Section: Discussionmentioning

confidence: 99%

“…In addition to the flexibility of dealing with both discontinuity types, DE‐FEM inherits all of the virtues of IGFEM/HIFEM: The construction of both weak and strong enrichment functions is straightforward, as they are based on the standard Lagrange shape functions of integration elements; Because enrichment functions vanish at original mesh nodes, the Kronecker‐delta property is maintained in standard nodes, allowing essential boundary conditions to be applied in the same way as in standard FEM; With the use of a diagonal preconditioner, or a proper scaling factor for the enrichment functions, the formulation used for treating weak discontinuities is stable, ie, the condition number increases at the same rate as that of standard FEM under mesh refinement; A hierarchical implementation of DE‐FEM can analyze multiple discontinuities and n ‐junctions within a single element. Cracks and interfaces are allowed to intersect each other as well …”

Section: Introductionmentioning

confidence: 99%

A stable interface‐enriched formulation for immersed domains with strong enforcement of essential boundary conditions

Boom

Zhang

Keulen

et al. 2019

Numerical Meth Engineering

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Aims and ScopeThe International Journal for Numerical Methods in Engineering publishes original papers describing significant, novel developments in numerical methods that are applicable to engineering problems. The Journal is known for welcoming contributions in a wide range of areas in computational engineering, including computational issues in model reduction, uncertainty quantification, verification and validation, inverse analysis and stochastic methods, optimisation, element technology, solution techniques and parallel computing, damage and fracture, mechanics at micro and nano-scales, low-speed fluid dynamics, fluid-structure interaction, electromagnetics, coupled diffusion phenomena, and error estimation and mesh generation. It is emphasized that this is by no means an exhaustive list, and particularly papers on multi-scale, multi-physics or multi-disciplinary problems, and on new, emerging topics are welcome. Manuscripts should have sufficient original numerical content, and generate new knowledge that is applicable to general classes of engineering problems, and not be limited to applications of existing methods, or propose incremental improvements to existing methods. The journal publishes full-length papers, which should normally be less than 25 journal pages in length, and short communications, which can be at most 8 journal pages. Discussions of papers in print can be published, but two-part papers will not be considered for review.

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“…where the first term on the right-hand-side of (7) corresponds to the traditional interpolation based on the shape functions and field values at the M nonconforming nodes, and the second term denotes the enriched component of the solution associated with the Q weakly (C 0 ) or strongly (C −1 ) discontinuous enrichment functions Ψ j and generalized degrees of freedom f j . 41 Beyond the simplicity of the meshing process, the main advantages of adopting the IGFEM in this multiscale shape optimization study resides in the stationary nature of the underlying nonconforming mesh, which simplifies the computation of the shape sensitivities and eliminates issues associated with mesh distortion. 16,[42][43][44] When C −1 enrichments are used to capture the strong discontinuity along the cohesive interfaces, every enrichment node has a corresponding "mirror node" tied by the cohesive failure law, as shown in Figure 2 for the case of a linear tetrahedral element traversed by a single material interface.…”

Section: Interface-enriched Generalized Finite Element Methodsmentioning

confidence: 99%

Multiscale design of three‐dimensional nonlinear composites using an interface‐enriched generalized finite element method

Brandyberry

Najafi

Geubelle

2020

Numerical Meth Engineering

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A computational framework is developed to model and optimize the nonlinear multiscale response of three-dimensional particulate composites using an interface-enriched generalized finite element method. The material nonlinearities are associated with interfacial debonding of inclusions from a surrounding matrix which is modeled using C −1 continuous enrichment functions and a cohesive failure model. Analytic material and shape sensitivities of the homogenized constitutive response are derived and used to drive a nonlinear inverse homogenization problem using gradient-based optimization methods. Spherical and ellipsoidal particulate microstructures are designed to match a component of the homogenized stress-strain response to a desired constructed macroscopic stress-strain behavior. K E Y W O R D Sanalytic sensitivity, cohesive failure, composites, multiscale modeling, shape optimization 1 2806

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A stable discontinuity-enriched finite element method for 3-D problems containing weak and strong discontinuities

Cited by 28 publications

References 45 publications

An interface-enriched generalized finite element method for level set-based topology optimization

An interface-enriched generalized finite element method for level set-based topology optimization

A stable interface‐enriched formulation for immersed domains with strong enforcement of essential boundary conditions

Multiscale design of three‐dimensional nonlinear composites using an interface‐enriched generalized finite element method

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