# Ted Belytschko's research while affiliated with Northwestern University and other places

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## Publications (408)

The upper internal structures which are located above the core and below the head cover may play a significant role in the hydrodynamics of an energy excursion such as a hypothetical core disruptive accident (HCDA) by mitigating the slug impact on the head. For purposes of studying these effects, a structural model was developed for the upper inter...

We present a regularized phenomenological multiscale model where elastic properties are computed using direct homogenization and subsequently evolved using a simple three-parameter orthotropic continuum damage model. The salient feature of the model is a unified regularization framework based on the concept of effective softening strain. The unifie...

An element-free Galerkin (EFG) method with linear, quadratic and cubic approximations, which can exactly, in a numerical sense, pass the corresponding patch tests is proposed and is named as consistent EFG (CEFG) method. The development of this method is based on the Hu–Washizu three-field variational principle. Numerical integration schemes with c...

This updated and expanded edition of the bestselling textbook provides a comprehensive introduction to the methods and theory of nonlinear finite element analysis. New material provides a concise introduction to some of the cutting-edge methods that have evolved in recent years in the field of nonlinear finite element modeling, and includes the eXt...

The implosion of an underwater structure is a dynamic event caused by the ambient external pressure. It produces a short duration pressure pulse that radiates outwards and can damage adjacent structures. This paper presents results from a combined experimental/numerical study that aims to understand the underlying physics and establish the paramete...

Ce travail propose d'investiguer l'extension des méthodes XFEM [1][2] à l'approche Isogéo-métrique [3][4]. Après présentation du cas test utilisé et discussion des méthodes permettant d'imposer convenablement les conditions aux limites, nous présentons les conditions permettant une convergence optimale quel que soit le degré des fonctions NURBS uti...

A method for representing discontinuous material properties in a heterogeneous domain by the extended finite element method (XFEM) has been applied to study ultrasonic wave propagation in polymer matrix particulate/fibrous composites. Representative volume elements of the composite material microstructure were generated by the random sequential ads...

We propose a modular approach for generalized computational mechanics in mesostructured continua, namely the archetype blending continuum (ABC) theory. The theory’s modularity derives from its mathematical constructors: archetypes, or building blocks that all multi-component material laws are generated from. These archetypes are the means used to d...

New procedures for modeling interactions among dislocations and nanosized cracks within the dynamically evolving bridging domain method (DEBDM) have been developed. The DEBDM is an efficient concurrent atomistic-tocontinuum approach based on the bridging domain method, where the atomic domain dynamically adapts to encompass evolving defects. New al...

An adaptive atomistic‐to‐continuum method is presented for modeling the propagation of material defects. This method extends the bridging domain method to allow the atomic domain to dynamically conform to the evolving defect regions during a simulation, without introducing spurious oscillations and without requiring mesh refinement. The atomic doma...

The consistency condition for the nodal derivatives in traditional meshfree Galerkin methods is only the differentiation of the approximation consistency (DAC). One missing part is the consistency between a nodal shape function and its derivatives in terms of the divergence theorem in numerical forms. In this paper, a consistency framework for the...

An anisotropic strain energy function is proposed for tensile loading in graphene that provides a nonlinear, hyperelastic constitutive equation. In the proposed function, the energy depends on the principal invariants of the right Cauchy–Green tensor and the strains in the zigzag and armchair directions. The use of the zigzag and armchair strains g...

En 1999, une extension de la méthode des éléments finis a été proposée. Baptisée depuis X-FEM (“eXtended Finite Element Method”), cette extension permet de modéliser des surfaces de discontinuité (fissures, interfaces matériaux, bords libres, …) sur un maillage sans que ce dernier doive s'y conformer. Ce papier dresse le bilan des avancées réalisée...

A new technique is presented to study fracture in nanomaterials by coupling quantum mechanics (QM) and continuum mechanics (CM). A key new feature of this method is that broken bonds are identified by a sharp decrease in electron density at the bond midpoint in the QM model. As fracture occurs, the crack tip position and crack path are updated from...

The extended finite element method (X-FEM) has proven to be an accurate, robust method for solving problems in fracture mechanics. X-FEM has typically been used with elements using linear basis functions, although some work has been performed using quadratics. In the current work, the X-FEM formulation is incorporated into isogeometric analysis to...

Single Point Incremental forming (SPIF) has generated significant interest recently due to its increased formability and greater process flexibility. However, the complicated deformation mechanisms involved in SPIF have prevented conclusive identification of the primary mechanisms responsible for failure. This work successfully predicts the forming...

A new method for modeling discontinuities, such as cracks, in the element free Galerkin method is presented. A jump function
is used for the displacement discontinuity along the crack faces and the Westergard’s solution enrichment near the crack tip.
These enrichments, being extrinsic, can be limited only to the nodes surrounding the crack. The met...

This paper presents a computational analysis and several simulations of an existing experiment, which deals with a quasi-static
thermal crack propagation in a glass plate. The experimental observation was that a straight or oscillatory crack propagation
occurred depending on the plate width and thermal loading. The goal here is to simulate this exp...

A continuum-atomistic adaptive multiscale method is developed for the simulation of the dynamics of dislocations. Two key features of the method are (i) methods for both refining and coarsening the model, where coarsening refers to a continuum to atomistic transition and refinement to the opposite and (ii) error criteria for refining and coarsening...

This paper presents a higher-order method for modeling dislocations with the extended finite element method (XFEM). This method is applicable to complex geometries, interfaces with lattice mismatch strains, and both anisotropic and spatially non-uniform material properties. A numerical procedure for computing the J-integral around a dislocation cor...

Incremental forming is a sheet metal forming process that has envisioned considerable interest in the research community due to greater formability, economical and product independent tooling and greater process flexibility. However, lack of the ability to predict fracture has considerably hindered its industrial adoption. This work uses finite ele...

Fast methods for determining the onset of instability for elastic–plastic damage models under multiaxial loading conditions are developed. On the basis of the general Hadamard instability criterion, we derive a closed-form expression to determine the onset of instability, the bifurcation directions and the polarization vector. The results of the pr...

Numerical and experimental studies on the impact and penetration of armor plates subjected to sub-ordnance range impact velocities by pointed and flat strikers is presented. Three target scenarios are considered: (1) a blank (unbacked) steel plate; (2) the same steel plate backed by a thick layer of polyurea; and (3) two identical steel plates of h...

An overview of the extended/generalized finite element method (GEFM/XFEM) with emphasis on methodological issues is presented. This method enables the accurate approximation of solutions that involve jumps, kinks, singularities, and other locally non-smooth features within elements. This is achieved by enriching the polynomial approximation space o...

Improvements in numerical aspects of dynamic crack propagation procedures by the extended finite element method are described and studied. Using only the discontinuous enrichment function in XFEM gives a binary description of the crack tip element: it is either cut or not. We describe a correction force to modify the forces to smoothly release the...

An explicit–explicit staggered time-integration algorithm and an implicit–explicit counterpart are presented for the solution of non-linear transient fluid–structure interaction problems in the Arbitrary Lagrangian–Eulerian (ALE) setting. In the explicit–explicit case where the usually desirable simultaneous updating of the fluid and structural sta...

A three-dimensional extended finite element method (X-FEM) coupled with a narrow band fast marching method (FMM) is developed and implemented in the Abaqus finite element package for curvilinear fatigue crack growth and life prediction analysis of metallic structures. Given the level set representation of arbitrary crack geometry, the narrow band F...

Multiscale methods are quickly becoming a new paradigm in many branches of science and engineering. This includes computational engineering, and to underline the growing importance of the subject the Editors of the International Journal for Numerical Methods in Engineering have decided to publish a Special Issue on Multiscale Computational Methods...

Many of the formulations of current research interest, including iosogeometric methods and the extended finite element method, use nontraditional basis functions. Some, such as subdivision surfaces, may not have convenient analytical representations. The concept of an element, if appropriate at all, no longer coincides with the traditional definiti...

The enrichment of the extended finite element method (XFEM) by meshfree approximations is studied. The XFEM allows for modeling
arbitrary discontinuities, but with low order elements the accuracy often needs improvement. Here, the meshfree approximation
is used as an enrichment in a cluster of nodes about the crack tip to improve accuracy. Several...

Finite element methods
(FEM) have been widely used in simulating the single point incremental forming (SPIF) process to investigate the effects of process parameters, such as incremental depth, tool size and tool path on the thickness/strain distributions, deformed shapes, and the formability. However, due to the complexity of the process and the...

We study several enrichment strategies for dynamic crack propagation in the context of the extended finite element method
and the effect of different directional criteria on the crack path. A new enrichment method with a time dependent enrichment
function is proposed. In contrast to previous approaches, it entails only one crack tip enrichment func...

An approximate level set method for three-dimensional crack propagation is presented. In this method, the discontinuity surface in each cracked element is defined by element-local level sets (ELLSs). The local level sets are generated by a fitting procedure that meets the fracture directionality and its continuity with the adjacent element crack su...

The onset and recurrence of cancer is one of the major biomedical quandaries of our time. Currently, surgically removed tumors
often leave behind a residual cancer cell population. As not all cancer cells can be detected to ensure complete tumor removal,
systemic and widespread chemotherapy is usually injected into the bloodstream to attempt to ta...

This paper deals with numerical crack propagation and makes use of the extended finite element method in the case of explicit dynamics. The advantage of this method is the absence of remeshing. The use of XFEM with Heaviside functions only gives a binary description of the crack tip element: cut or not. Here, we modify the internal forces with a co...

This paper deals with numerical crack propagation and makes use of the extended finite element method in the case of explicit dynamics. The advantage of this method is the absence of remeshing. The use of XFEM with Heaviside functions only gives a binary descrip- tion of the crack tip element: cut or not. Here, we modify the internal forces with a...

lation or in the vicinity of the crack tip 14. In related work, Armero and Ehrlich 15 used embedded discontinuity elements to model hinge lines in plates. The development of a fracture criterion that is computationally efficient and is easily applied in terms of available data poses a significant difficulty. Fracture criteria for quasibrittle mater...

The application of a multiscale method, called the multiscale aggregating discontinuities (MAD) method, to the failure analysis of composites is described. Two distinct features of the MAD method are the use of perforated unit cells, and the extraction of coarse-grained failure information. In the perforated unit cell, all subdomains of the unit ce...

Understanding the nanoscale fracture mechanisms is critical for tailoring the mechanical properties of materials at small length scales. We perform an atomistic study to characterize the formation and extension of nano-sized cracks. By using atomistic reaction pathway calculations, we determine the energetics governing the brittle and ductile respo...

Dislocation models based on the extended finite element method (XFEM) are developed for thin shells such as carbon nanotubes (CNTs) and thin films. In shells, methods for edge dislocations, which move by glide, and prismatic dislocations, which move by climb, are described. In thin films, methods for dislocations with edge, screw and/or prismatic c...

A method for the modeling of dislocations and cracks by atomistic/continuum models is described. The methodology combines the extended finite element method with the bridging domain method (BDM). The former is used to model crack surfaces and slip planes in the continuum, whereas the BDM is used to link the atomistic models with the continuum. The...

The extended and generalized flnite element methods are reviewed with an emphasis on their applications to problems in material science: (1) fracture (2) dislocations (3) grain boundaries and (4) phases interfaces. These methods facilitate the modeling of complicated geometries and the evolution of such geometries, particularly when combined with l...

Recent studies have finally produced accurate measurements of the mechanical properties of carbon nanotubes, confirming the
anticipated results computed from quantum and molecular mechanics. Several studies have also predicted an enhancement of these
material properties as a result of electron irradiation. Here we prove conclusively through a rigor...

Crack instabilities and the phenomenon of crack speed saturation in a brittle material (PMMA) are studied with a meshfree cracking particle method. We reproduce the experimental observation that the computed terminal crack speeds attained in PMMA specimens are substantially lower than the Rayleigh wave speed; the computed crack speeds agree quite w...

A bridging domain method for coupled atomistic–continuum models is proposed that enables to compare various coupling terms. The approach does not require the finite element mesh to match the lattice spacing of the atomic model. It is based on an overlapping domain decomposition method that makes use of Lagrange multipliers and weight functions in t...

Stabilized stress-point integration schemes based on gradient stabilization and dilatational stabilization methods are presented for linear elastostaticity problems in the framework of element-free Galerkin (EFG) method. The instability in stress fields associated with the stress-point integration is treated by the addition to the Galerkin weak for...

A new method for modeling discrete cracks based on the extended finite element method is described. In the method, the growth of the actual crack is tracked and approximated with contiguous discrete crack segments that lie on finite element nodes and span only two adjacent elements. The method can deal with complicated fracture patterns because it...

The bridging domain method is an overlapping domain decomposition approach for coupling finite element continuum models and molecular mechanics models. In this method, the total energy is decomposed into atomistic and continuum parts by complementary weight functions applied to each part of the energy in the coupling domain. To enforce compatibilit...

The fracture of tetrahedral amorphous carbon (ta-C) with sp3 fraction 60% is investigated at the nano-scale with molecular dynamics simulations using the Environment-Dependent Interatomic
Potential [1], and compared with fracture of diamond. It is found that the tetrahedral amorphous carbon fractures very differently from
diamond, and these fractur...

A method for coarse graining of microcrack growth to the macroscale through the multiscale aggregating discontinuity (MAD) method is further developed. Three new features are: (1) methods for treating nucleating cracks, (2) the linking of the micro unit cell with the macroelement by the hourglass mode, and (3) methods for recovering macrocracks wit...

A new support integration technique is proposed, which is similar to those used in truly mesh-free methods. The contribution of this paper is to exploit the divergence-free condition for the support integrals to construct quadrature formulas that only require three integration points per particle in two dimensions. Numerical examples show that the...

A discontinuous-Galerkin method for large deformation fluid–structure interaction problems is developed. The fluid–structure interface can be arbitrarily aligned relative to the fluid grid. An Eulerian description is used for the fluid with a Lagrangian description of the solid. Results are presented for several examples that show excellent agreeme...

Two issues in the extended finite element method (XFEM) are addressed: efficient numerical integration of the weak form when the enrichment function is self-equilibrating and blending of the enrichment. The integration is based on transforming the domain integrals in the weak form into equivalent contour integrals. It is shown that the contour form...

A method for treating fluid–structure interaction of fracturing structures under impulsive loads is described. The coupling method is simple and does not require any modifications when the structure fails and allows fluid to flow through openings between crack surfaces. Both the fluid and the structure are treated by meshfree methods. For the struc...

Stabilized stress-point integration schemes based on Least-Squares Stabilization (LSS), Taylor series Expansion Based Stabilization
(TEBS) and Finite Increment Gradient (FIG) are compared for linear elastostaticity problems and some relations between them
are described. Particular emphasis is placed on stress-point integration procedures with stabi...

An overlaid domain decomposition method, called the bridging domain method, for the coupled simulation of molecular/continuum systems is analyzed. In this method, compatibility between the atomistic and the continuum subdomains is enforced by Lagrange multipliers. Two forms are considered: (1) a consistent constraint form, which employs the exact n...

The performance of finite element methods for dynamic crack propagation in brittle materials is studied. Three methods are
considered: the extended finite element method (XFEM), element deletion method and interelement crack method. The extended
finite element method is a method for arbitrary crack propagation without remeshing. In element deletion...

Strategies for coupling quantum mechanical (QM), molecular mechanical (MM), and continuum mechanical (CM) methods are described. For QM/MM coupling, we consider two overlapping domain schemes: the widely-used ONIOM method that involves full overlap, and a new minimal-overlap scheme denoted as the “quantum to molecular mechanical overlapping domain”...

Empirical potentials that are commonly used in molecular mechanical (MM) calculations often exhibit marked differences from quantum mechanical (QM) calculations. These differences can lead to mismatches in the mechanical properties of different subdomains in coupled QM/MM calculations that can result in artifactual behavior or low accuracy. We pres...

In the extended finite element method (XFEM), errors are caused by parasitic terms in the approximation space of the blending elements at the edge of the enriched subdomain. A discontinuous Galerkin (DG) formulation is developed, which circumvents this source of error. A patch-based version of the DG formulation is developed, which decomposes the d...

Stress-point integration provides significant reductions in the computational effort of mesh-free Galerkin methods by using fewer integration points, and thus facilitates the use of mesh-free methods in applications where full integration would be prohibitively expensive. The influence of stress-point integration on the convergence and stability pr...

A weak form and an implementation are given for fluid–structure interaction by the immersed/fictitious element method for compressible fluids. The weak form is applicable to models where the fluid is described by Eulerian coordinates while the solid is described by Lagrangian coordinates, which suits their intrinsic characteristics. A unique featur...

An adaptive method within the extended finite element method (XFEM) framework which adapts the enrichment function locally to the physics of a problem, as opposed to polynomial or mesh refinement, is presented. The method minimizes a local residual and determines the parameters of the enrichment function. We consider an energy form and a ‘strong’ f...

New methods for the analysis of failure by multiscale methods that invoke unit cells to obtain the subscale response are described. These methods, called multiscale aggregating discontinuities, are based on the concept of ‘perforated’ unit cells, which exclude subdomains that are unstable, i.e. exhibit loss of material stability. Using this concept...

A recently developed finite element method for the modeling of dislocations is improved by adding enrichments in the neighborhood of the dislocation core. In this method, the dislocation is modeled by a line or surface of discontinuity in two or three dimensions. The method is applicable to nonlinear and anisotropic materials, large deformations, a...

The fracture of tetrahedral amorphous carbon at the nanoscale was investigated with molecular dynamics simulations using the environment-dependent interatomic potential. It was found that the fracture strength of amorphous carbon nanospecimens is insensitive to initial cracks with diameters smaller than about 40 A, i.e., the material exhibits flaw...

Microsystems have become an integral part of our lives and can be found in homeland security, medical science, aerospace applications
and beyond. Many critical microsystem applications are in harsh environments, in which long-term reliability needs to be guaranteed
and repair is not feasible. For example, gyroscope microsystems on satellites need t...

In this presentation we will discuss a continuum multiscale framework which combines the Bridging Domain Method (BDM) of Xiao and Belytschko [1] with the eXtended Finite Element Method (XFEM) of of Moës et al. [2]. The BDM is a hierarchical overlapping domain decomposition scheme. Material in the coarse-scale domain is modelled as a continuum using...

A Monte Carlo based scheme for the formation of graphite oxide (GO) was developed and implemented. A Rosenbluth factor was used to select intermediate structures in an attempt to form stable, low-energy, and realistic GO. The scheme resulted in the production of GO that has an interplanar spacing of 5.8 Å, in good agreement with the experimental va...

A method for the analysis of shear bands using local partition of unity is developed in the framework of the extended finite element method (XFEM). Enrichments are introduced for both the displacement field and the thermal field. The shear band width is determined by minimizing the plastic work. A coupled finite strain thermo-elastoplastic constitu...