Showing papers in "Computer Methods in Applied Mechanics and Engineering in 2016"
TL;DR: A new computing paradigm is developed, which is referred to as data-driven computing, according to which calculations are carried out directly from experimental material data and pertinent constraints and conservation laws, thus bypassing the empirical material modeling step of conventional computing altogether.
Abstract: We develop a new computing paradigm, which we refer to as data-driven computing, according to which calculations are carried out directly from experimental material data and pertinent constraints and conservation laws, such as compatibility and equilibrium, thus bypassing the empirical material modeling step of conventional computing altogether. Data-driven solvers seek to assign to each material point the state from a prespecified data set that is closest to satisfying the conservation laws. Equivalently, data-driven solvers aim to find the state satisfying the conservation laws that is closest to the data set. The resulting data-driven problem thus consists of the minimization of a distance function to the data set in phase space subject to constraints introduced by the conservation laws. We motivate the data-driven paradigm and investigate the performance of data-driven solvers by means of two examples of application, namely, the static equilibrium of nonlinear three-dimensional trusses and linear elasticity. In these tests, the data-driven solvers exhibit good convergence properties both with respect to the number of data points and with regard to local data assignment. The variational structure of the data-driven problem also renders it amenable to analysis. We show that, as the data set approximates increasingly closely a classical material law in phase space, the data-driven solutions converge to the classical solution. We also illustrate the robustness of data-driven solvers with respect to spatial discretization. In particular, we show that the data-driven solutions of finite-element discretizations of linear elasticity converge jointly with respect to mesh size and approximation by the data set.
458 citations
TL;DR: In this paper, a cubic degradation function was proposed to provide a stress-strain response prior to crack initiation, which more closely approximates linear elastic behavior, and a derivation of the governing equations in terms of a general energy potential from balance laws that describe the kinematics of both the body and phase-field.
Abstract: Phase-field models have been a topic of much research in recent years. Results have shown that these models are able to produce complex crack patterns in both two and three dimensions. A number of extensions from brittle to ductile materials have been proposed and results are promising. To date, however, these extensions have not accurately represented strains after crack initiation or included important aspects of ductile fracture such as stress triaxiality. This work introduces a number of contributions to further develop phase-field models for fracture in ductile materials. These contributions include: a cubic degradation function that provides a stress–strain response prior to crack initiation that more closely approximates linear elastic behavior, a derivation of the governing equations in terms of a general energy potential from balance laws that describe the kinematics of both the body and phase-field, introduction of a yield surface degradation function that provides a mechanism for plastic softening and corrects the non-physical elastic deformations after crack initiation, a proposed mechanism for including a measure of stress triaxiality as a driving force for crack initiation and propagation, and a correction to an error in the configuration update of an elastoplastic return-mapping algorithm for J 2 flow theory. We also present a heuristic time stepping scheme that facilitates computations that require a relatively long load time prior to crack initiation. A number of numerical results will be presented that demonstrate the effects of these contributions.
383 citations
TL;DR: In this article, a nonintrusive projection-based model reduction approach for full models based on time-dependent partial differential equations is presented, which is applicable to full models that are linear in the state or have a low-order polynomial nonlinear term.
Abstract: This work presents a nonintrusive projection-based model reduction approach for full models based on time-dependent partial differential equations. Projection-based model reduction constructs the operators of a reduced model by projecting the equations of the full model onto a reduced space. Traditionally, this projection is intrusive, which means that the full-model operators are required either explicitly in an assembled form or implicitly through a routine that returns the action of the operators on a given vector; however, in many situations the full model is given as a black box that computes trajectories of the full-model states and outputs for given initial conditions and inputs, but does not provide the full-model operators. Our nonintrusive operator inference approach infers approximations of the reduced operators from the initial conditions, inputs, trajectories of the states, and outputs of the full model, without requiring the full-model operators. Our operator inference is applicable to full models that are linear in the state or have a low-order polynomial nonlinear term. The inferred operators are the solution of a least-squares problem and converge, with sufficient state trajectory data, in the Frobenius norm to the reduced operators that would be obtained via an intrusive projection of the full-model operators. Our numerical results demonstrate operator inference on a linear climate model and on a tubular reactor model with a polynomial nonlinear term of third order.
307 citations
TL;DR: A mechanistic, data-driven, two-scale approach is developed for predicting the behavior of general heterogeneous materials under irreversible processes such as inelastic deformation and is believed to open new avenues in parameter-free multi-scale modeling of complex materials, and perhaps in other fields that require homogenization of irreversible processes.
Abstract: The discovery of efficient and accurate descriptions for the macroscopic behavior of materials with complex microstructure is an outstanding challenge in mechanics of materials. A mechanistic, data-driven, two-scale approach is developed for predicting the behavior of general heterogeneous materials under irreversible processes such as inelastic deformation. The proposed approach includes two major innovations: (1) the use of a data compression algorithm, k -means clustering, during the offline stage of the method to homogenize the local features of the material microstructure into a group of clusters; and (2) a new method called self-consistent clustering analysis used in the online stage that is valid for any local plasticity laws of each material phase without the need for additional calibration. A particularly important feature of the proposed approach is that the offline stage only uses the linear elastic properties of each material phase, making it efficient. This work is believed to open new avenues in parameter-free multi-scale modeling of complex materials, and perhaps in other fields that require homogenization of irreversible processes.
287 citations
TL;DR: In this article, a macroscopic framework is proposed for a continuum phase field modeling of fracture in porous media, which provides a rigorous geometric approach to a diffusive crack modeling based on the introduction of a constitutive balance equation for a regularized crack surface and its modular linkage to a Darcy-Biot type bulk response of hydro-poro-elasticity.
Abstract: The prediction of fluid- and moisture-driven crack propagation in deforming porous media has achieved increasing interest in recent years, in particular with regard to the modeling of hydraulic fracturing , the so-called “ fracking ”. Here, the challenge is to link at least three modeling ingredients for (i) the behavior of the solid skeleton and fluid bulk phases and their interaction, (ii) the crack propagation on not a priori known paths and (iii) the extra fluid flow within developed cracks. To this end, a macroscopic framework is proposed for a continuum phase field modeling of fracture in porous media. It provides a rigorous geometric approach to a diffusive crack modeling based on the introduction of a constitutive balance equation for a regularized crack surface and its modular linkage to a Darcy–Biot-type bulk response of hydro-poro-elasticity. The approach overcomes difficulties associated with the computational realization of sharp crack discontinuities, in particular when it comes to complex crack topologies including branching. A modular concept is outlined for linking of the diffusive crack modeling with the hydro-poro-elastic response of the porous bulk material. This includes a generalization of crack driving forces from energetic definitions towards threshold-based criteria in terms of the effective stress related to the solid skeleton of a fluid-saturated porous medium. Furthermore, a Poiseuille-type constitutive continuum modeling of the extra fluid flow in developed cracks is suggested based on a deformation-dependent permeability, that is scaled by a characteristic length. This proposed modular model structure is exploited in the numerical implementation by constructing a robust finite element method, based on an algorithmic decoupling of updates for the crack phase field and the state variables of the hydro-poro-elastic bulk response. We demonstrate the performance of the phase field formulation of fracture for a spectrum of model problems of hydraulic fracture. A slight modification of the framework allows the simulation of drying-caused crack patterns in partially saturated capillar-porous media.
279 citations
TL;DR: Robust and efficient numerical algorithms for pressure-driven and fluid-driven settings in which the focus relies on mesh adaptivity in order to save computational cost for large-scale 3D applications are developed.
Abstract: This work presents phase field fracture modeling in heterogeneous porous media. We develop robust and efficient numerical algorithms for pressure-driven and fluid-driven settings in which the focus relies on mesh adaptivity in order to save computational cost for large-scale 3D applications. In the fluid-driven framework, we solve for three unknowns pressure, displacements and phase field that are treated with a fixed-stress iteration in which the pressure and the displacement–phase-field system are decoupled. The latter subsystem is solved with a combined Newton approach employing a primal–dual active set method in order to account for crack irreversibility. Numerical examples for pressurized fractures and fluid filled fracture propagation in heterogeneous porous media demonstrate our developments. In particular, mesh refinement allows us to perform systematic studies with respect to the spatial discretization parameter.
235 citations
TL;DR: In this paper, a phase-field model of crack regularization was proposed for elastic and elasto-plastic materials, where two independent phase fields correspond to the lower and upper faces of the shell.
Abstract: With the theme of fracture of finite-strain plates and shells based on a phase-field model of crack regularization, we introduce a new staggered algorithm for elastic and elasto-plastic materials. To account for correct fracture behavior in bending, two independent phase-fields are used, corresponding to the lower and upper faces of the shell. This is shown to provide a realistic behavior in bending-dominated problems, here illustrated in classical beam and plate problems. Finite strain behavior for both elastic and elasto-plastic constitutive laws is made compatible with the phase-field model by use of a consistent updated-Lagrangian algorithm. To guarantee sufficient resolution in the definition of the crack paths, a local remeshing algorithm based on the phase-field values at the lower and upper shell faces is introduced. In this local remeshing algorithm, two stages are used: edge-based element subdivision and node repositioning. Five representative numerical examples are shown, consisting of a bi-clamped beam, two versions of a square plate, the Keesecker pressurized cylinder problem, the Hexcan problem and the Muscat-Fenech and Atkins plate. All problems were successfully solved and the proposed solution was found to be robust and efficient.
234 citations
TL;DR: In this article, an explicit topology optimization approach based on moving morphable components with curved skeletons (central lines) is proposed, which is achieved by constructing the topology description function (TDF) which describes the geometry of a structural component with curved skeleton explicitly in an elegant way.
Abstract: In the present paper, an explicit topology optimization approach based on moving morphable components (MMC) with curved skeletons (central lines) is proposed. This is achieved by constructing the topology description function (TDF) which describes the geometry of a structural component with curved skeleton explicitly in an elegant way. The proposed method has very flexible geometry modeling capability and represents a substantial improvement of the geometry modeling capability of existing MMC based approaches. Numerical examples demonstrate the effectiveness of the proposed approach.
232 citations
TL;DR: In this paper, a smoothed displacement jump approximation is introduced by means of level-set functions to overcome the issue of pixelized interfaces in voxel-based models, which allows interaction between bulk and interface cracks, and for arbitrary geometries and interactions between cracks.
Abstract: In this work, a formulation is developed within the phase field method for mod-eling interactions between interfacial damage and bulk brittle cracking in complex microstructures. The method is dedicated to voxel-based models of highly complex microstructures, as obtained from X-ray microtomography images. A smoothed displacement jump approximation is introduced by means of level-set functions to overcome the issue of pixelized interfaces in voxel-based models. A simple technique is proposed to construct the level-set function in that case. Compared to recent work aiming at modeling cohesive cracks within the phase field method, our framework differs in several points: the formulation is such that interfaces are not initially damaged; no additional variables are required to describe the discontinuities at the interface and fatigue cracks can be modeled. The technique allows interaction between bulk and interface cracks, e.g. nucleation from interfaces and propagation within the matrix, and for arbitrary geometries and interactions between cracks. Several benchmarks are presented to validate the model. The technique is illustrated through numerical examples involving complex microcracking in X-ray CT image-based models of concrete microstructures.
222 citations
TL;DR: In this paper, the peridynamic equations of motion were recast by recasting Navier's displacement equilibrium equations into their nonlocal form by introducing the per-idynamic differential operator, which permits the non-local form of expressions for the determination of the stress and strain components.
Abstract: The nonlocal peridynamic theory has been proven extremely robust for predicting damage nucleation and propagation in materials under complex loading conditions. Its equations of motion, originally derived based on the principle of virtual work, do not contain any spatial derivatives of the displacement components. Thus, their solution does not require special treatment in the presence of geometric and material discontinuities. This study presents an alternative approach to derive the peridynamic equations of motion by recasting Navier’s displacement equilibrium equations into their nonlocal form by introducing the peridynamic differential operator. Also, this operator permits the nonlocal form of expressions for the determination of the stress and strain components. The capability of this differential operator is demonstrated by constructing solutions to ordinary, partial differential equations and derivatives of scattered data, as well as image compression and recovery without employing any special filtering and regularization techniques.
219 citations
TL;DR: In this article, the diffusion equation for the phase-field can be conceived as a special case of the averaging equation of a gradient-damage model where the damage is averaged, and subtle differences in the degradation functions commonly adopted in damage and phase field approaches are key to the observation that the fracture process zone does not broaden in the wake of the crack tip.
Abstract: Gradient-enhanced damage models and phase-field models are seemingly very disparate approaches to fracture Whereas gradient-enhanced damage models find their roots in damage mechanics, which is a smeared approach from the onset, and gradients were added to restore well-posedness beyond a critical strain level, the phase-field approach to brittle fracture departs from a discontinuous description of failure, where the distribution function is regularised, leading to the inclusion of spatial gradients as well Herein, we will consider both approaches, and discuss their similarities and differences The averaging (diffusion) equations for the averaging field and the phase-field will be compared, and it is shown that the diffusion equation for the phase-field can be conceived as a special case of the averaging equation of a gradient-damage model where the damage is averaged Further, the role of the driving force is examined, and it is shown that subtle differences in the degradation functions commonly adopted in damage and phase-field approaches are key to the observation that, different from damage mechanics, the fracture process zone does not broaden in the wake of the crack tip
TL;DR: An efficient discretization method for the solution of the unsteady incompressible Navier–Stokes equations based on a high order (Hybrid) Discontinuous Galerkin formulation is presented and the performance on two and three dimensional benchmark problems is demonstrated.
Abstract: In this paper we present an efficient discretization method for the solution of the unsteady incompressible Navier–Stokes equations based on a high order (Hybrid) Discontinuous Galerkin formulation. The crucial component for the efficiency of the discretization method is the distinction between stiff linear parts and less stiff non-linear parts with respect to their temporal and spatial treatment. Exploiting the flexibility of operator-splitting time integration schemes we combine two spatial discretizations which are tailored for two simpler sub-problems: a corresponding hyperbolic transport problem and an unsteady Stokes problem. For the hyperbolic transport problem a spatial discretization with an Upwind Discontinuous Galerkin method and an explicit treatment in the time integration scheme is rather natural and allows for an efficient implementation. The treatment of the Stokes part involves the solution of linear systems. In this case a discretization with Hybrid Discontinuous Galerkin methods is better suited. We consider such a discretization for the Stokes part with two important features: H ( div ) -conforming finite elements to guarantee exactly divergence-free velocity solutions and a projection operator which reduces the number of globally coupled unknowns. We present the method, discuss implementational aspects and demonstrate the performance on two and three dimensional benchmark problems.
TL;DR: This work proposes a faster and equally accurate approach for quasi-static phase-field computing of (brittle) fracture using a monolithic solution scheme which is accompanied by a novel line search procedure to overcome the iterative convergence issues of non-convex minimization.
Abstract: Phase-field modeling of fracture phenomena in solids is a very promising approach which has gained popularity within the last decade. However, within the finite element framework, already a two-dimensional quasi-static phase-field formulation is computationally quite demanding, mainly for the following reasons: (i) the need to resolve the small length scale inherent to the diffusive crack approximation calls for extremely fine meshes, at least locally in the crack phase-field transition zone, (ii) due to non-convexity of the related free-energy functional, a robust, but slowly converging staggered solution scheme based on algorithmic decoupling is typically used. In this contribution we tackle problem (ii) and propose a faster and equally accurate approach for quasi-static phase-field computing of (brittle) fracture using a monolithic solution scheme which is accompanied by a novel line search procedure to overcome the iterative convergence issues of non-convex minimization. We present a detailed critical evaluation of the approach and its comparison with the staggered scheme in terms of computational cost, accuracy and robustness.
TL;DR: In this article, a modified peridynamic correspondence material model was proposed to avoid zero-energy mode instability in a peridynamics particle code, where a term was added to the correspondence strain energy density that resists deviations from a uniform deformation.
Abstract: Peridynamic correspondence material models provide a way to combine a material model from the local theory with the inherent capabilities of peridynamics to model long-range forces and fracture. However, correspondence models in a typical particle discretization suffer from zero-energy mode instability. These instabilities are shown here to be an aspect of material stability. A stability condition is derived for state-based materials starting from the requirement of potential energy minimization. It is shown that all correspondence materials fail this stability condition due to zero-energy deformation modes of the family. To eliminate these modes, a term is added to the correspondence strain energy density that resists deviations from a uniform deformation. The resulting material model satisfies the stability condition while effectively leaving the stress tensor unchanged. Computational examples demonstrate the effectiveness of the modified material model in avoiding zero-energy mode instability in a peridynamic particle code.
TL;DR: This review paper addresses the so called geometric multiscale approach for the numerical simulation of blood flow problems, from its origin (that the authors can collocate in the second half of '90s) to their days, and details the most popular numerical algorithms for the solution of the coupled problems.
Abstract: This review paper addresses the so called geometric multiscale approach for the numerical simulation of blood flow problems, from its origin (that we can collocate in the second half of '90s) to our days. By this approach the blood fluid-dynamics in the whole circulatory system is described mathematically by means of heterogeneous problems featuring different degree of detail and different geometric dimension that interact together through appropriate interface coupling conditions. Our review starts with the introduction of the stand-alone problems, namely the 3D fluid-structure interaction problem, its reduced representation by means of 1D models, and the so-called lumped parameters (aka 0D) models, where only the dependence on time survives. We then address specific methods for stand-alone 3D models when the available boundary data are not enough to ensure the mathematical well posedness. These so-called "defective problems" naturally arise in practical applications of clinical relevance but also because of the interface coupling of heterogeneous problems that are generated by the geometric multiscale process. We also describe specific issues related to the boundary treatment of reduced models, particularly relevant to the geometric multiscale coupling. Next, we detail the most popular numerical algorithms for the solution of the coupled problems. Finally, we review some of the most representative works-from different research groups-which addressed the geometric multiscale approach in the past years. A proper treatment of the different scales relevant to the hemodynamics and their interplay is essential for the accuracy of numerical simulations and eventually for their clinical impact. This paper aims at providing a state-of-the-art picture of these topics, where the gap between theory and practice demands rigorous mathematical models to be reliably filled. (C) 2016 Elsevier B.V. All rights reserved.
TL;DR: Dalcin et al. as mentioned in this paper presented a study on metodos computacionales in the context of the CONICET project of the Centro Cientifico Tecnologico Conicet.
Abstract: Fil: Dalcin, Lisandro Daniel. King Abdullah University of Science and Technology; Arabia Saudita. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Centro Cientifico Tecnologico Conicet - Santa Fe. Centro de Investigaciones en Metodos Computacionales. Universidad Nacional del Litoral. Centro de Investigaciones en Metodos Computacionales; Argentina
TL;DR: A new class of unfitted finite element methods with high order accurate numerical integration over curved surfaces and volumes which are only implicitly defined by level set functions is introduced.
Abstract: We introduce a new class of unfitted finite element methods with high order accurate numerical integration over curved surfaces and volumes which are only implicitly defined by level set functions. An unfitted finite element method which is suitable for the case of piecewise planar interfaces is combined with a parametric mapping of the underlying mesh resulting in an isoparametric unfitted finite element method. The parametric mapping is constructed in a way such that the quality of the piecewise planar interface reconstruction is significantly improved allowing for high order accurate computations of (unfitted) domain and surface integrals. We present the method, discuss implementational aspects and present numerical examples which demonstrate the quality and potential of this method.
TL;DR: In this paper, a postbuckling analysis of carbon nanotube (CNT) reinforced functionally graded plates with edges elastically restrained against translation and rotation is presented, to the best of our knowledge.
Abstract: This paper presents, to the authors’ knowledge, a first known postbuckling analysis of carbon nanotube (CNT) reinforced functionally graded plates with edges elastically restrained against translation and rotation. The plate considered is of moderate thickness and, hence, the first-order shear deformation theory (FSDT) and von Karman assumption are adopted to incorporate the effects of transverse shear strains, rotary inertia and moderate rotations. The element-free IMLS-Ritz method is employed. The cubic spline weight function and linear basis are utilized in the approximation. In this study, the bending stiffness is evaluated through the nodal integration scheme. The postbuckling path is traced using the arc-length method combined with the modified Newton–Raphson technique. Parametric studies on the postbuckling behavior of CNT reinforced functionally graded plates are conducted to examine the effects of CNT content by volume, plate width-to-thickness ratio and plate aspect ratio by varying the elastically restrained parameters of translation and rotation on the boundaries. The results of the present study are obtained for simplified cases so that comparison studies can be undertaken using the values reported in the literature.
TL;DR: This paper analyzes several iterative solution schemes for solving matrix systems that result from discretization and linearization of the governing equations and highlights the fundamental connections that underlie their effectiveness.
Abstract: Coupled poromechanical problems appear in a variety of disciplines, from reservoir engineering to biomedical applications. This work focuses on efficient strategies for solving the matrix systems that result from discretization and linearization of the governing equations. These systems have an inherent block structure due to the coupled nature of the mass and momentum balance equations. Recently, several iterative solution schemes have been proposed that exhibit stable and rapid convergence to the coupled solution. These schemes appear distinct, but a unifying feature is that they exploit the block-partitioned nature of the problem to accelerate convergence. This paper analyzes several of these schemes and highlights the fundamental connections that underlie their effectiveness. We begin by focusing on two specific methods: a fully-implicit and a sequential-implicit scheme. In the first approach, the system matrix is treated monolithically, and a Krylov iteration is used to update pressure and displacement unknowns simultaneously. To accelerate convergence, a preconditioning operator is introduced based on an approximate block-factorization of the linear system. Next, we analyze a sequential-implicit scheme based on the fixed-stress split. In this method, one iterates back and forth between updating displacement and pressure unknowns separately until convergence to the coupled solution is reached. We re-interpret this scheme as a block-preconditioned Richardson iteration, and we show that the preconditioning operator is identical to that used within the fully-implicit approach. Rapid convergence in both the Richardson- and Krylov-based methods results from a particular choice for a sparse Schur complement approximation. This analysis leads to a unified framework for developing solution schemes based on approximate block factorizations. Many classic fully-implicit and sequential-implicit schemes are simple sub-cases. The analysis also highlights several new approaches that have not been previously explored. For illustration, we directly compare the performance and robustness of several variants on a benchmark problem.
TL;DR: In this article, a Reduced Order Model (ROM) was proposed for CFD applications based on Finite Volume approximation, starting from the results available in turbulent Reynold-Averaged Navier-Stokes simulations in order to enlarge the application field of Proper Orthogonal Decomposition-Reduced Order Models (POD-ROM) technique to more industrial fields.
Abstract: Numerical simulation of fluid flows requires important computational efforts but it is essential in engineering applications. Reduced Order Model (ROM) can be employed whenever fast simulations are required, or in general, whenever a trade-off between computational cost and solution accuracy is a preeminent issue as in process optimization and control. In this work, the efforts have been put to develop a ROM for Computational Fluid Dynamics (CFD) application based on Finite Volume approximation, starting from the results available in turbulent Reynold-Averaged Navier–Stokes simulations in order to enlarge the application field of Proper Orthogonal Decomposition-Reduced Order Model (POD-ROM) technique to more industrial fields. The approach is tested in the classic benchmark of the numerical simulation of the 2D lid-driven cavity. In particular, two simulations at Re=103 and Re=105 have been considered in order to assess both a laminar and a turbulent case. Some quantities have been compared with the Full Order Model in order to assess the performance of the proposed ROM procedure i.e., the kinetic energy of the system and the reconstructed quantities of interest (velocity, pressure and turbulent viscosity). In addition, for the laminar case, the comparison between the ROM steady-state solution and the data available in literature has been presented. The results have turned out to be very satisfactory both for the accuracy and the computational times. As a major outcome, the approach turns out not to be affected by the energy blow up issue characterizing the results obtained by classic turbulent POD-Galerkin methods.
TL;DR: In this article, the phase field is defined as a two-dimensional field on the midsurface of the structure and the variation of strains through the shell thickness is considered and the split into tensile and compressive elastic energy components, needed to prevent cracking in compression, has to be carried out at various points through the thickness.
Abstract: We present an approach for phase-field modeling of fracture in thin structures like plates and shells, where the kinematics is defined by midsurface variables. Accordingly, the phase field is defined as a two-dimensional field on the midsurface of the structure. In this work, we consider brittle fracture and a Kirchhoff–Love shell model for structural analysis. We show that, for a correct description of fracture, the variation of strains through the shell thickness has to be considered and the split into tensile and compressive elastic energy components, needed to prevent cracking in compression, has to be carried out at various points through the thickness, which prohibits the typical separation of the elastic energy into membrane and bending terms. For numerical analysis, we employ isogeometric discretizations and a rotation-free Kirchhoff–Love shell formulation. In several numerical examples we show the applicability of the approach and detailed comparisons with 3D solid simulations confirm its accuracy and efficiency.
TL;DR: In this article, a coupled continuous/discontinuous approach is proposed to model the two failure phases of quasi-brittle materials in a coherent way, which involves an integral-type nonlocal continuum damage model coupled with an extrinsic discrete interface model.
Abstract: Failure of quasi-brittle materials is governed by crack formation and propagation which can be characterized by two phases: (i) diffuse material degradation process with initial crack formation and (ii) severe localization of damage leading to the propagation of large cracks and fracture. While continuum damage mechanics provides an excellent framework to describe the first failure phase, it is unable to represent discontinuous displacement fields. In sharp contrast, cohesive zone models are poorly suited for describing diffuse damage but can accurately resolve discrete cracks. In this manuscript, we propose a coupled continuous/discontinuous approach to model the two failure phases of quasi-brittle materials in a coherent way. The proposed approach involves an integral-type nonlocal continuum damage model coupled with an extrinsic discrete interface model. The transition from diffuse damage to macroscopic cohesive cracks is made through an equivalent thermodynamic framework established in multidimensional settings, in which the dissipated energy is computed numerically and weakly matched. The method is implemented within the extended finite element framework, which allows for crack propagation without remeshing. A few benchmark problems involving straight and curved cracks are investigated to demonstrate the applicability and robustness of the coupled XFEM cohesive-damage approach. Force–displacement responses, as well as predicted propagation paths, are presented and shown to be in close agreement with available experimental data. Furthermore, the method is found to be insensitive to various damage threshold values for damage–crack transition, yielding energetically consistent results.
TL;DR: In this article, an efficient multi-material topology optimization strategy for seeking the optimal layout of structures considering the cohesive constitutive relationship of the interface is presented. But, the interface behavior may exhibit tension/compression non-symmetric topology, in which material interfaces mainly undergo compression.
Abstract: In most of the existing topology optimization studies of multi-material structures, the interface of different materials was assumed to be perfectly bonded. Optimal design based on the perfect-interface assumption may introduce the risk of failure caused by interface debonding. This paper presents an efficient multi-material topology optimization strategy for seeking the optimal layout of structures considering the cohesive constitutive relationship of the interface. Based on the color level set method to describe the topology and the interface, the interface behavior is simulated by combining the extended finite element method (XFEM) and the cohesive model on fixed mesh. This enables modeling of possible separation of material interfaces, and thus provides a more realistic model of multi-material structures. Furthermore, this interface modeling technique avoids the difficulty of re-meshing when tracking the moving cohesive interface positions during the optimization process. In the topology optimization model, the normal velocities defined on the level set points are considered as design variables. In conjunction with the adjoint-variable sensitivity analysis, these design variables are updated by using the mathematical programming approach and then used to interpolate the boundary velocities. These boundary velocities are extrapolated to the whole domain with the fast marching method and used to advance the structural boundary through the Hamilton–Jacobi equation. This topology optimization technique can handle multiple constraints easily in the framework of level set method and at the same time preserve the signed distance property of the level set functions. Two numerical examples are given to demonstrate the effectiveness of the present method. It is also revealed that the optimal design considering interface behavior may exhibit tension/compression non-symmetric topology, in which material interfaces mainly undergo compression.
TL;DR: In this article, a composite truncated conical shell with embedded single-walled carbon nanotubes (SWCNTs) subjected to an external pressure and axial compression simultaneously is considered.
Abstract: The present research deals with bifurcation and vibration responses of a composite truncated conical shell with embedded single-walled carbon nanotubes (SWCNTs) subjected to an external pressure and axial compression simultaneously. The distribution of reinforcements through the thickness of the shell is assumed to be either uniform or functionally graded. The equations of motion are established using Green–Lagrange type nonlinear kinematics within the framework of Novozhilov nonlinear shell theory. Linear membrane prebuckling analysis is conducted to extract the prebuckling deformations. The stability equations are derived by applying the adjacent equilibrium criterion to the prebuckling state of the conical shell. A semi-analytical solution on the basis of the trigonometric expansion through the circumferential direction along with the harmonic differential quadrature (HDQ) discretization in the meridional direction is developed. A series of comparison studies are carried out to assure the accuracy and the convergence of the HDQ method. The research indicates that the superb accuracy and efficiency of solutions with few grid points are attributed to the higher-order harmonic approximation function in the HDQ method. Parametric studies are also presented to investigate the influence of boundary conditions, semi-vertex angle of the cone, volume fraction and distribution of CNTs on stability and vibration characteristics of the truncated conical shell. The results show that both volume fraction and distribution of CNTs play a pivotal role in the natural frequencies, buckling mode and buckling loads of the FG-CNTRC truncated conical shell.
TL;DR: In this article, a local/nonlocal coupling technique called the morphing method is developed to couple classical continuum mechanics with state-based peridynamics, which enables the description of cracks that appear and propagate spontaneously, is applied to the key domain of a structure.
Abstract: A local/nonlocal coupling technique called the morphing method is developed to couple classical continuum mechanics with state-based peridynamics. State-based peridynamics, which enables the description of cracks that appear and propagate spontaneously, is applied to the key domain of a structure, where damage and fracture are considered to have non-negligible effects. In the rest of the structure, classical continuum mechanics is used to reduce computational costs and to simultaneously satisfy solution accuracy and boundary conditions. Both models are glued by the proposed morphing method in the transition region. The morphing method creates a balance between the stiffness tensors of classical continuum mechanics and the weighted coefficients of state-based peridynamics through the equivalent energy density of both models. Linearization of state-based peridynamics is derived by Taylor approximations based on vector operations. The discrete formulation of coupled models is also described. Two-dimensional numerical examples illustrate the validity and accuracy of the proposed technique. It is shown that the morphing method, originally developed for bond-based peridynamics, can be successfully extended to state-based peridynamics through the original developments presented here.
TL;DR: In this article, a similarity transformation is used to convert the governing momentum and energy equations into non-linear ordinary differential equations with the appropriate boundary conditions, which are solved analytically by Duan-Rach approach.
Abstract: In this paper, the unsteady squeezing flow and heat transfer of MHD nanofluid between the infinite parallel plates with thermal radiation effect is investigated. A similarity transformation is used to convert the governing momentum and energy equations into non-linear ordinary differential equations with the appropriate boundary conditions. These non-linear ordinary differential equations are solved analytically by Duan–Rach Approach ( DRA ). This method allows us to find a solution without using numerical methods to evaluate the undetermined coefficients. This method modifies the standard Adomian Decomposition Method ( ADM ) by evaluating the inverse operators at the boundary conditions directly. The effects of various parameters such as the squeeze number, the magnetic parameter, the volume fraction of nanofluid, the Eckert number and the radiation parameter are investigated on the velocity and temperature. Also, the values of skin friction coefficient and the Nusselt number are calculated and presented through figures. The results show that the temperature profile and Nusselt number increase with the increase of radiation parameter. Furthermore, the limiting cases are obtained and are found to be in good agreement with the previously published results.
TL;DR: Numerical examples demonstrate that the method is able to deliver highly accurate domain integrals with a minimal number of quadrature points and provides a viable alternative to standard octree-based approaches when combined with the Finite Cell Method.
Abstract: This paper presents an efficient and accurate method for the integration of discontinuous functions on a background mesh in three dimensions. This task is important in computational mechanics applications where internal interfaces are present in the computational domain. The proposed method creates boundary-conforming integration subcells for composed numerical quadrature, even in the presence of sharp geometric features (e.g. edges or vertices). Similar to the octree procedure, the algorithm subdivides the cut elements into eight octants. However, the octant nodes are moved onto the interface, which allows for a robust resolution of the intersection topology in the element while maintaining algorithmic simplicity. Numerical examples demonstrate that the method is able to deliver highly accurate domain integrals with a minimal number of quadrature points. Further examples show that the proposed method provides a viable alternative to standard octree-based approaches when combined with the Finite Cell Method.
TL;DR: By exploiting a multilevel control structure, truncated hierarchical B-spline representations support interactive modeling tools, while simultaneously providing effective approximation schemes for the manipulation of complex data sets and the solution of partial differential equations via isogeometric analysis.
Abstract: Local refinement with hierarchical B-spline structures is an active topic of research in the context of geometric modeling and isogeometric analysis. By exploiting a multilevel control structure, we show that truncated hierarchical B-spline (THB-spline) representations support interactive modeling tools, while simultaneously providing effective approximation schemes for the manipulation of complex data sets and the solution of partial differential equations via isogeometric analysis. A selection of illustrative 2D and 3D numerical examples demonstrates the potential of the hierarchical framework.
TL;DR: In this article, the authors investigated fracture in shells with a phase-field modeling approach, which is based on solid-shell kinematics with small rotations and displacements and is discretized using quadratic non-uniform rational B-spline basis functions.
Abstract: In this paper, we investigate fracture in shells with a phase-field modeling approach. The shell model is based on solid-shell kinematics with small rotations and displacements and is discretized using quadratic Non-Uniform Rational B-Spline basis functions. Membrane and shear locking is alleviated through the Assumed Natural Strain approach. The solid-shell formulation is combined with a brittle phase-field model for elastic materials, as well as with a ductile fracture model for elasto-plastic materials exhibiting J 2 plasticity with isotropic hardening. Several examples demonstrate the ability of the proposed framework to capture crack initiation, propagation, merging and branching phenomena as well as crack bulging effects in shells under different states of loading.
TL;DR: This is the first study on layout design of multiple engineering features using level-set functions (LSFs) and Boolean operations and numerical examples are tested to demonstrate the validity and merits of the proposed feature-driven topology optimization for complicated design problems.
Abstract: In this paper, a feature-driven topology optimization method is developed. This is the first study on layout design of multiple engineering features using level-set functions (LSFs) and Boolean operations. The novelty of this work is threefold. First, multiple engineering features of arbitrary shape are considered as basic design primitives and topology variation is achieved via the layout and shape optimization of the features. Kreisselmeier–Steinhauser (KS) function constructed by means of Boolean operations is adopted as the LSF, which uses an implicit function to ensure a smooth description and topological changes of basic features and the whole structure. Second, using a modified Heaviside function to smooth the void–solid material transition over a fixed computing mesh, a narrow-band domain integral scheme is developed for the efficient sensitivity analysis. Third, the gray material distribution regions at the feature-connecting portions are analyzed and the underlying reason for that is traced to the non-equidistant distribution of level-set contours of specific features. To avoid the gray regions, an approximated signed distance function is proposed to regularize the LSF and KS function. The bounded normalization property of the KS function is highlighted for its construction with the signed distance functions or normalized first-order approximations. Numerical examples are finally tested to demonstrate the validity and merits of the proposed feature-driven topology optimization for complicated design problems.