Showing papers in "Computer Methods in Applied Mechanics and Engineering in 2015"
TL;DR: In this paper, a generalization of recently developed continuum phase field models for brittle fracture towards fully coupled thermo-mechanical and multi-physics problems at large strains is presented.
Abstract: This work presents a generalization of recently developed continuum phase field models for brittle fracture towards fully coupled thermo-mechanical and multi-physics problems at large strains. It outlines a rigorous geometric approach to the diffusive crack modeling based on the introduction of a balance of regularized crack surface, governed by a crack phase field. The regularized crack surface functional is based on a crack surface density function, that describes the macroscopic crack surface in the bulk material per unit of the reference volume. The approach overcomes difficulties associated with the computational realization of sharp crack discontinuities, in particular when it comes to complex crack topologies. The formulation proposed is essentially a gradient damage theory, however, equipped with critical ingredients rooted in fracture mechanics. A modular concept is outlined for the linking of the diffusive crack modeling with complex multi-field response of the bulk material, where focus is put on the model problem of finite thermo-elasticity. This concerns a generalization of crack driving forces from the energetic definitions towards stress-based criteria, the constitutive modeling of heat conduction across cracks and convective heat exchanges at crack faces based on additional constitutive functions. This is achieved by approximating surface load integrals of the sharp crack approach by distinct volume integrals. We demonstrate the performance of the phase field formulation of fracture at large strains by means of representative numerical examples.
416 citations
TL;DR: In this article, a generalization of recently developed continuum phase field models from brittle to ductile fracture coupled with thermo-plasticity at finite strains is presented, which uses a geometric approach to the diffusive crack modeling based on the introduction of a balance equation for a regularized crack surface.
Abstract: This work presents a generalization of recently developed continuum phase field models from brittle to ductile fracture coupled with thermo-plasticity at finite strains. It uses a geometric approach to the diffusive crack modeling based on the introduction of a balance equation for a regularized crack surface and its modular linkage to a multi-physics bulk response developed in the first part of this work. This evolution equation is governed by a constitutive crack driving force. In this work, we supplement the energetic and stress-based forces for brittle fracture by additional forces for ductile fracture. These are related to state variables associated with the inelastic response, such as the amount of plastic strain and the void volume fraction in metals, or the amount of craze strains in glassy polymers. To this end, we define driving forces based on elastic and plastic work densities , and barrier functions related to critical values of these inelastic state variables. The proposed thermodynamically consistent framework of ductile phase field fracture is embedded into a formulation of gradient thermo-plasticity, that is able to account for material length scales such as the width of shear bands. It is applied to two constitutive model problems. The first is designed for the analysis of brittle-to-ductile failure mode transition in the dynamic failure analysis of metals . The second is constructed for a quasi-static analysis of crazing-induced fracture in glassy polymers . A spectrum of simulations demonstrates that the use of barrier-type crack driving forces in the phase field modeling of fracture, governed by accumulated plastic strains in metals or crazing strains in polymers, provide results in very good agreement with experiments.
407 citations
TL;DR: This paper develops a geometrically flexible technique for computational fluid-structure interaction (FSI) that directly analyzes a spline-based surface representation of the structure by immersing it into a non-boundary-fitted discretization of the surrounding fluid domain, and introduces the term "immersogeometric analysis" to identify this paradigm.
Abstract: In this paper, we develop a geometrically flexible technique for computational fluid–structure interaction (FSI). The motivating application is the simulation of tri-leaflet bioprosthetic heart valve function over the complete cardiac cycle. Due to the complex motion of the heart valve leaflets, the fluid domain undergoes large deformations, including changes of topology. The proposed method directly analyzes a spline-based surface representation of the structure by immersing it into a non-boundary-fitted discretization of the surrounding fluid domain. This places our method within an emerging class of computational techniques that aim to capture geometry on non-boundary-fitted analysis meshes. We introduce the term “immersogeometric analysis” to identify this paradigm.
The framework starts with an augmented Lagrangian formulation for FSI that enforces kinematic constraints with a combination of Lagrange multipliers and penalty forces. For immersed volumetric objects, we formally eliminate the multiplier field by substituting a fluid–structure interface traction, arriving at Nitsche’s method for enforcing Dirichlet boundary conditions on object surfaces. For immersed thin shell structures modeled geometrically as surfaces, the tractions from opposite sides cancel due to the continuity of the background fluid solution space, leaving a penalty method. Application to a bioprosthetic heart valve, where there is a large pressure jump across the leaflets, reveals shortcomings of the penalty approach. To counteract steep pressure gradients through the structure without the conditioning problems that accompany strong penalty forces, we resurrect the Lagrange multiplier field. Further, since the fluid discretization is not tailored to the structure geometry, there is a significant error in the approximation of pressure discontinuities across the shell. This error becomes especially troublesome in residual-based stabilized methods for incompressible flow, leading to problematic compressibility at practical levels of refinement. We modify existing stabilized methods to improve performance.
To evaluate the accuracy of the proposed methods, we test them on benchmark problems and compare the results with those of established boundary-fitted techniques. Finally, we simulate the coupling of the bioprosthetic heart valve and the surrounding blood flow under physiological conditions, demonstrating the effectiveness of the proposed techniques in practical computations.
390 citations
TL;DR: An extended isogeometric element formulation (XIGA) for analysis of through-the-thickness cracks in thin shell structures is developed in this article, where the discretization is based on Non-Uniform Rational B-Splines (NURBS).
Abstract: An extended isogeometric element formulation (XIGA) for analysis of through-the-thickness cracks in thin shell structures is developed. The discretization is based on Non-Uniform Rational B-Splines (NURBS). The proposed XIGA formulation can reproduce the singular field near the crack tip and the discontinuities across the crack. It is based on the Kirchhoff–Love theory where C 1 -continuity of the displacement field is required. This condition is satisfied by the NURBS basis functions. Hence, the formulation eliminates the need of rotational degrees of freedom or the discretization of the director field facilitating the enrichment strategy. The performance and validity of the formulation is tested by several benchmark examples.
320 citations
TL;DR: In this article, an arbitrary-order locking-free method for linear elasticity is proposed, which relies on a pure-displacement (primal) formulation and leads to a symmetric, positive definite system matrix with compact stencil.
Abstract: We devise an arbitrary-order locking-free method for linear elasticity The method relies on a pure-displacement (primal) formulation and leads to a symmetric, positive definite system matrix with compact stencil The degrees of freedom are vector-valued polynomials of arbitrary order k ⩾ 1 on the mesh faces, so that in three space dimensions, the lowest-order scheme only requires 9 degrees of freedom per mesh face The method can be deployed on general polyhedral meshes The key idea is to reconstruct the symmetric gradient and divergence inside each mesh cell in terms of the degrees of freedom by solving inexpensive local problems The discrete problem is assembled cell-wise using these operators and a high-order stabilization bilinear form Locking-free error estimates are derived for the energy norm and for the L 2 -norm of the displacement, with optimal convergence rates of order ( k + 1 ) and ( k + 2 ) , respectively, for smooth solutions on general meshes The theoretical results are confirmed numerically, and the CPU cost is evaluated on both standard and polygonal meshes
320 citations
TL;DR: In this article, force convection heat transfer in a lid driven semi annulus enclosure is studied in presence of non-uniform magnetic field and the calculations were performed for different governing parameters namely, the Reynolds number, nanoparticle volume fraction and Hartmann number.
Abstract: In this paper force convection heat transfer in a lid driven semi annulus enclosure is studied in presence of non-uniform magnetic field. The enclosure is filled with Fe3O4–water nanofluid. It is assumed that the magnetization of the fluid is varying linearly with temperature and magnetic field intensity. Control volume based finite element method is used to solve the governing equations in the form of vorticity–stream function formulation. The calculations were performed for different governing parameters namely, the Reynolds number, nanoparticle volume fraction and Hartmann number. Results show that Nusselt number has direct relationship with Reynolds number, nanoparticle volume fraction while it has reverse relationship with Hartmann number.
311 citations
TL;DR: In this paper, the effect of the squeeze number, nanofluid volume fraction, Hartmann number and heat source parameter on flow and heat transfer was investigated, and the results showed that skin friction coefficient increases with increase of the Nusselt number and Hartmann numbers but it decreases with an increase in the volume fraction.
Abstract: The problem of nanofluid hydrothermal behavior in presence of variable magnetic field is investigated analytically using Differential Transformation Method. The fluid in the enclosure is water containing different types of nanoparticles: Al2O3 and CuO. The effective thermal conductivity and viscosity of nanofluid are calculated by KKL (Koo–Kleinstreuer–Li) correlation. In this model effect of Brownian motion on the effective thermal conductivity is considered. The comparison between the results from Differential Transformation Method and previous work are in well agreement which proved the capability of this method for solving such problems. The effect of the squeeze number, nanofluid volume fraction, Hartmann number and heat source parameter on flow and heat transfer is investigated. The results show that skin friction coefficient increases with increase of the squeeze number and Hartmann number but it decreases with increase of nanofluid volume fraction. Nusselt number increases with augment of nanoparticle volume fraction, Hartmann number while it decreases with increase of the squeeze number.
311 citations
TL;DR: In this paper, a primal-dual active set strategy is proposed to enforce crack irreversibility as a constraint, which can be identified as a semi-smooth Newton method, and the active set iteration is merged with the Newton iteration for solving the fully-coupled nonlinear partial differential equation discretized using finite elements.
Abstract: In this paper, we consider phase-field based fracture propagation in elastic media. The main purpose is the development of a robust and efficient numerical scheme. To enforce crack irreversibility as a constraint, we use a primal-dual active set strategy, which can be identified as a semi-smooth Newton method. The active set iteration is merged with the Newton iteration for solving the fully-coupled nonlinear partial differential equation discretized using finite elements, resulting in a single, rapidly converging nonlinear scheme. It is well known that phase-field models require fine meshes to accurately capture the propagation dynamics of the crack. Because traditional estimators based on adaptive mesh refinement schemes are not appropriate, we develop a predictor-corrector scheme for local mesh adaptivity to reduce the computational cost. This method is both robust and efficient and allows us to treat temporal and spatial refinements and to study the influence of model regularization parameters. Finally, our proposed approach is substantiated with different numerical tests for crack propagation in elastic media and pressurized fracture propagation in homogeneous and heterogeneous media.
300 citations
TL;DR: A density-based topology optimization approach is proposed to design structures with strict minimum length scale based on using a filtering-threshold topology optimized scheme and computationally cheap geometric constraints.
Abstract: A density-based topology optimization approach is proposed to design structures with strict minimum length scale. The idea is based on using a filtering-threshold topology optimization scheme and computationally cheap geometric constraints. The constraints are defined over the underlying structural geometry represented by the filtered and physical fields. Satisfying the constraints leads to a design that possesses user-specified minimum length scale. Conventional topology optimization problems can be augmented with the proposed constraints to achieve minimum length scale on the final design. No additional finite element analysis is required for the constrained optimization. Several benchmark examples are presented to show the effectiveness of this approach.
266 citations
TL;DR: In this article, the authors present formulations for compressible and incompressible hyperelastic thin shells which can use general 3D constitutive models and enforce the necessary plane stress condition analytically for incompressibly and iteratively for compressibly.
Abstract: We present formulations for compressible and incompressible hyperelastic thin shells which can use general 3D constitutive models. The necessary plane stress condition is enforced analytically for incompressible materials and iteratively for compressible materials. The thickness stretch is statically condensed and the shell kinematics are completely described by the first and second fundamental forms of the midsurface. We use C 1 -continuous isogeometric discretizations to build the numerical models. Numerical tests, including structural dynamics simulations of a bioprosthetic heart valve, show the good performance and applicability of the presented methods.
254 citations
TL;DR: In this article, a differentiable geometry projection is proposed for the continuous topology optimization of linearly elastic planar structures made of bars of fixed width and semicircular ends, where the out-of-plane thickness is penalized so that the optimizer is capable of removing bars during the optimization.
Abstract: This article describes a method for the continuum-based topology optimization of structures made of discrete elements. In particular, we examine the optimization of linearly elastic planar structures made of bars of fixed width and semicircular ends. The design space for the optimization consists of the endpoint locations of the bar’s medial axes and their out-of-plane thicknesses. To circumvent re-meshing upon design changes, we project the design onto a fixed analysis grid using a differentiable geometry projection that results in a density field indicating the fraction of solid material anywhere in the design space, as in density-based topology optimization methods. The out-of-plane thickness is penalized so that the optimizer is capable of removing bars during the optimization. The differentiability of the projection allows for the computation via the chain rule of design sensitivities of responses of interest, and therefore it allows for the use of robust and efficient gradient-based optimization methods. Notably, this approach makes it easier to fabricate optimal designs by using off-the-shelf stock material. Furthermore, the method considers the case where bars overlap at their joints (i.e. their thicknesses are added at the joint) and when they do not. Finally, our proposed method naturally accommodates the imposition of several fixed length scales. We demonstrate the proposed approach on classical problems of compliance-based topology optimization and identify its benefits as well as research directions to be addressed in the future.
TL;DR: In this article, the authors proposed a new Multi-Material Level Set (MM-LS) topology description model for topology and shape optimization of structures involving multiple materials, where each phase is represented by a combined formulation of different level set functions.
Abstract: This paper proposes a new Multi-Material Level Set (MM-LS) topology description model for topology and shape optimization of structures involving multiple materials. Each phase is represented by a combined formulation of different level set functions. With a total number of M level set functions, this approach provides a representation of M materials and one void phase (totally M + 1 phases). The advantages of the proposed method include: (1) it can guarantee that each point contains exactly one phase, without overlaps between each two phases and redundant regions within the design domain; (2) it possesses an explicit mathematical expression, which greatly facilitates the design sensitivity analysis; and (3) it retains the merits of the level set method, including smooth boundary and distinct interface. A parametric level set method is applied to evolve the topology and shape of multi-material structures, with a high computational efficiency. Several numerical examples are presented to demonstrate the effectiveness of the proposed method.
TL;DR: A new concept called analysis in computer aided design (AiCAD) is proposed for design-through-analysis workflow that uses non-uniform rational B-Splines (NURBS)-based B-Rep models for the entire workflow.
Abstract: A new concept called analysis in computer aided design (AiCAD) is proposed for design-through-analysis workflow. This concept uses non-uniform rational B-Splines (NURBS)-based B-Rep models for the entire workflow. Such models consist of trimmed NURBS surfaces and are considered standard in the industry, especially for modeling free-form geometries. The newly developed isogeometric B-Rep analysis (IBRA) used in AiCAD is also presented. IBRA can be considered as a generalization of isogeometric analysis (IGA) that uses the boundary representation (B-Rep) of the design model in addition to the same basis functions as in IGA for approximating the solution fields. IBRA provides the framework for creating a direct and complete analysis model from computer aided design (CAD) in a consistent finite-element-like manner. Thus, IBRA allows analyzing a CAD model without remodeling and meshing, even for complex geometries. For the numerical integration of trimmed surfaces, the concept of nested Jacobian approach (NEJA) with NURBS surfaces is introduced. In addition, for enforcing the different types of boundary conditions or mechanical entities, a new finite element type called isogeometric B-Rep element is introduced. Elements of this type permit enforcing, e.g., coupling or Dirichlet boundary conditions. A corresponding formulation based on a penalty approach is presented as well. The proposed workflow is realized exemplarily for surface modeling and the geometrical nonlinear analysis of shell structures. The differences between the standard analysis procedure and the AiCAD workflow are explained in detail. Various numerical examples confirm the accuracy, flexibility, and robustness of the proposed IBRA concept, thus highlighting its advantages for the realization of design-through-analysis workflow with a uniform geometry representation.
TL;DR: In this article, an algorithm that combines the computation of the Floquet exponents with bordering techniques is developed for the detection and tracking of bifurcations of nonlinear systems.
Abstract: The harmonic balance (HB) method is widely used in the literature for analyzing the periodic solutions of nonlinear mechanical systems. The objective of this paper is to exploit the method for bifurcation analysis, i.e., for the detection and tracking of bifurcations of nonlinear systems. To this end, an algorithm that combines the computation of the Floquet exponents with bordering techniques is developed. A new procedure for the tracking of Neimark–Sacker bifurcations that exploits the properties of eigenvalue derivatives is also proposed. The HB method is demonstrated using numerical experiments of a spacecraft structure that possesses a nonlinear vibration isolation device.
TL;DR: In this paper, the Virtual Element Method (VEM) is proposed for nonlinear elastic and inelastic problems, mainly focusing on a small deformation regime, and the numerical scheme is based on a low-order approximation of the displacement field.
Abstract: We present a Virtual Element Method (VEM) for possibly nonlinear elastic and inelastic problems, mainly focusing on a small deformation regime. The numerical scheme is based on a low-order approximation of the displacement field, as well as a suitable treatment of the displacement gradient. The proposed method allows for general polygonal and polyhedral meshes, it is efficient in terms of number of applications of the constitutive law, and it can make use of any standard black-box constitutive law algorithm. Some theoretical results have been developed for the elastic case. Several numerical results within the 2D setting are presented, and a brief discussion on the extension to large deformation problems is included.
TL;DR: The application of mortar methods in the framework of isogeometric analysis is investigated theoretically as well as numerically in this article, where two choices of uniformly stable spaces are presented, both of them are spline spaces but of a different degree.
Abstract: The application of mortar methods in the framework of isogeometric analysis is investigated theoretically as well as numerically. For the Lagrange multiplier two choices of uniformly stable spaces are presented, both of them are spline spaces but of a different degree. In one case, we consider an equal order pairing for which a cross point modification based on a local degree reduction is required. In the other case, the degree of the dual space is reduced by two compared to the primal. This pairing is proven to be inf–sup stable without any necessary cross point modification. Several numerical examples confirm the theoretical results and illustrate additional aspects.
TL;DR: This paper considers a plate in bending where the dynamic reduced-order model quickly adapts to changes in structural properties and achieves speedups of four orders of magnitude compared to rebuilding a model from scratch.
Abstract: Data-driven model reduction constructs reduced-order models of large-scale systems by learning the system response characteristics from data. Existing methods build the reduced-order models in a computationally expensive offline phase and then use them in an online phase to provide fast predictions of the system. In cases where the underlying system properties are not static but undergo dynamic changes, repeating the offline phase after each system change to rebuild the reduced-order model from scratch forfeits the savings gained in the online phase. This paper proposes dynamic reduced-order models that break with this classical but rigid approach. Dynamic reduced-order models exploit the opportunity presented by dynamic sensor data and adaptively incorporate sensor data during the online phase. This permits online adaptation to system changes while circumventing the expensive rebuilding of the model. A computationally cheap adaptation is achieved by constructing low-rank updates to the reduced operators. With these updates and with sufficient and accurate data, our approach recovers the same model that would be obtained by rebuilding from scratch. We demonstrate dynamic reduced-order models on a structural assessment example in the context of real-time decision making. We consider a plate in bending where the dynamic reduced-order model quickly adapts to changes in structural properties and achieves speedups of four orders of magnitude compared to rebuilding a model from scratch.
TL;DR: In this article, an efficient computational approach based on refined plate theory including the thickness stretching effect, namely quasi-3D theory, in conjunction with isogeometric formulation (IGA), is proposed for the size-dependent bending, free vibration and buckling analysis of functionally graded nanoplate structures.
Abstract: In this paper, an efficient computational approach based on refined plate theory (RPT) including the thickness stretching effect, namely quasi-3D theory, in conjunction with isogeometric formulation (IGA) is proposed for the size-dependent bending, free vibration and buckling analysis of functionally graded nanoplate structures. The present novel quasi-3D theory not only possesses 4 variables as refined plate theory but also accounts for both shear deformation and stretching effect without any requirement of shear correction factors (SCFs). The size-dependent effect is taken into account by nonlocal elasticity theory. Isogeometric analysis shows a great advantage in dealing with the high continuity and high order derivative requirements of the displacement fields used in quasi-3D and nonlocal theory. The reliability and accuracy of the present method are ascertained by comparing the obtained results with other published ones. Numerical examples are also performed to show the significance of nonlocal effect, material distribution profile, aspect ratios and boundary conditions on the behaviour of FGM nanoplates.
TL;DR: This article validation of Francfort and Marigo’s variational approach to fracture based on some classical fracture experiments shows that this approach can be used to faithfully account for unknown crack paths even for complex loadings and geometry.
Abstract: In this article, we focus on the validation of Francfort and Marigo’s variational approach to fracture based on some classical fracture experiments. We show that this approach can be used to faithfully account for unknown crack paths even for complex loadings and geometry. We revisit the backtracking algorithm, aimed at avoiding some spurious local minimizers of the total fracture energy and introduce a variant: the deep backtracking algorithm.
TL;DR: In this article, an extension of the extended isogeometric analysis (XIGA) for simulation of two-dimensional fracture mechanics problems in piezoelectric materials under dynamic and static coupled electromechanical loads is presented.
Abstract: Accurate numerical modeling of multifield piezoelectric materials is challenging because of the inherent electro-mechanical coupling effect and material anisotropic behaviors. The modeling becomes even more difficult especially for problems with non-smooth solutions like crack under dynamic loading. We present in this paper an extension of the extended isogeometric analysis (XIGA) for simulation of two-dimensional fracture mechanics problems in piezoelectric materials under dynamic and static coupled electromechanical loads. The discretization of problem domain is based on basis functions generated from NURBS, which are used for both geometric description and approximation of solution field variables. To capture the discontinuity across the crack-faces and the singularity at the crack-tip, the isogeometric approximation is locally enriched by discontinuous Heaviside function and asymptotic crack-tip branch functions. The sixfold enrichment functions particularly suitable for electromechanical crack-tip singularity of piezoelectric materials are used. To evaluate the generalized fracture parameters, a domain-form of electromechanical interaction integral is employed. For dynamic analysis, the implicit time integration scheme considering inertial effect is taken. Five numerical examples for single and mixed-modes of impermeable cracks are considered and the generalized fracture parameters under dynamic and static loads are analyzed. The accuracy and effectiveness of the proposed XIGA are illustrated through numerical investigations of the generalized dynamic and static fracture parameters. Numerical results are validated against the reference solutions derived from the boundary element methods. The effects of some numerical aspect ratios on generalized fracture parameters are also investigated. Additionally, we present some numerical results of quasi-static crack propagation in piezoelectric solids using the developed XIGA, taking fracture toughness anisotropy of polarized electroelastic materials into account, and employing the maximum modified hoop stress intensity factor criterion for predicting the growing direction of crack.
TL;DR: In this paper, a multiphase Smoothed Particle Hydrodynamics (SPH) method is applied to simulate the phenomena of bubbles rising and coalescing in three dimensions.
Abstract: The numerical simulation of bubbly flows is challenging due to the unstable multiphase interfaces with large density ratios and viscous ratios. The multiphase Smoothed Particle Hydrodynamics (SPH) method is applied in this paper to simulate the phenomena of bubbles rising and coalescing in three dimensions. Firstly, the multiphase SPH model is introduced in detail, including the derivation of the discretized governing equations based on the principle of virtual work, the viscous force, the multiphase interface treatment, the time-stepping scheme, the boundary implementation, etc. Considering the expensive computational cost in three-dimensional (3-D) SPH simulations, the effects of the scale of the computational domain and the density ratio on the multiphase interface are numerically investigated in order to decrease the amount of calculation. Afterwards, several cases of single bubbles rising through viscous fluids are tested and the SPH results are validated by both the experimental data and other numerical results in the literature. Furthermore, the phenomena of bubbles coalescing in both vertical and horizontal directions are simulated and the results agree well with the experimental data. It is found that the background pressure in the equation of state is essential to keep the multiphase interface smooth and stable when the Bond number is relatively small. The fair agreements between the results of SPH and other reference results demonstrate that the present multiphase SPH model is robust and stable enough to accurately simulate the dynamic phenomena of rising bubbles in different conditions, which can give a reference for engineering applications.
TL;DR: In this paper, a reduced database model is proposed for multiscale structural topology optimization, where at the microscopic scale, local materials are optimized concurrently according to current loading status.
Abstract: This paper builds on our recent work (Xia and Breitkopf, 2014) on multiscale structural topology optimization where at the microscopic scale, local materials are optimized concurrently according to current loading status. The former design framework requires intensive computational cost due to large number of repetitive local material optimizations. To circumvent this limitation, in the present work, we construct a reduced database model viewing the local material optimization process as a generalized constitutive behavior using separated representations. In this model, the database is built from a set of numerical experiments of local material optimizations in the macroscopic strain tensor space. Each value in the database corresponds to the strain energy density evaluated on a material microstructure, optimized according to the imposed macroscopic strain. By tensor decomposition, a continuous representation of the strain energy density is built as a sum of products of one dimensional interpolation functions. As a result of this a priori off-line step, the effective strain–energy and stress–strain relations required for macroscopic structural evaluation and optimization are provided in a numerically explicit manner. The results given by the reduced database model are compared with full-scale results. It is also shown that this explicit constitutive behavior representation can well serve multiscale structural design at a significantly reduced computational cost.
TL;DR: The versatility and accuracy of the proposed isogeometric boundary element method for problems in elasticity are demonstrated by convergence studies showing optimal rates and real world examples in two and three dimensions.
Abstract: An isogeometric boundary element method for problems in elasticity is presented, which is based on an independent approximation for the geometry, traction and displacement field This enables a flexible choice of refinement strategies, permits an efficient evaluation of geometry related information, a mixed collocation scheme which deals with discontinuous tractions along non-smooth boundaries and a significant reduction of the right hand side of the system of equations for common boundary conditions All these benefits are achieved without any loss of accuracy compared to conventional isogeometric formulations The system matrices are approximated by means of hierarchical matrices to reduce the computational complexity for large scale analysis For the required geometrical bisection of the domain, a strategy for the evaluation of bounding boxes containing the supports of NURBS basis functions is presented The versatility and accuracy of the proposed methodology are demonstrated by convergence studies showing optimal rates and real world examples in two and three dimensions
TL;DR: In this paper, a variationally consistent weak coupling method for thin-walled shell patches is proposed to ensure a corresponding geometric continuity in the deformed configuration and a correct transfer of bending moments across the interface.
Abstract: Thin shell structures are widely used in the aerospace, automotive and mechanical engineering industries. They are ideal candidates for the isogeometric analysis paradigm profiting from the smoothness of the geometry model, and the higher order approximation and higher continuity properties of NURBS. To model complex shell structures which need to be assembled from multiple patches, the bending stiffness should be maintained across the patch interfaces. We propose a variationally consistent weak coupling method for thin-walled shell patches. The method overcomes the need for C1-continuity along the patch interface to ensure a corresponding geometric continuity in the deformed configuration and a correct transfer of bending moments across the interface. Importantly, it allows a blended coupling of shells based on different mathematical models, e.g. Kirchhoff–Love and solid-like shell models. The proposed approach retains the high level of accuracy of single patch solutions and reveals its potential for authentic multi-patch NURBS modeling. We illustrate the good performance of the method for pure Kirchhoff–Love shell models and blended shell models with various examples. The presented approach supports local model refinements where e.g. full 3D stress states are of interest, and further opens the door for the coupling of laminated composites belonging to different lamina theories.
TL;DR: In this paper, a Markov Chain Monte Carlo (MCMC) sampling strategy was proposed for orthogonal polynomials of Hermite and Legendre types under each respective natural sampling distribution.
Abstract: Independent sampling of orthogonal polynomial bases via Monte Carlo is of interest for uncertainty quantification of models using Polynomial Chaos (PC) expansions. It is known that bounding the spectral radius of a random matrix consisting of PC samples yields a bound on the number of samples necessary to identify coefficients in the PC expansion via solution to a least squares regression problem. We present a related analysis which guarantees a mean square convergence using a coherence parameter of the sampled PC basis that may be both analytically bounded and computationally estimated. Utilizing asymptotic results for orthogonal polynomials, we bound the coherence parameter for polynomials of Hermite and Legendre type under each respective natural sampling distribution. In both polynomial bases we identify an importance sampling distribution which yields a bound with weaker dependence on the order of the PC basis. For more general orthonormal bases, we propose the coherence-optimal sampling: a Markov Chain Monte Carlo sampling, which directly uses the basis functions under consideration to achieve a statistical optimality among all such sampling schemes with identical support, and which guarantees recovery with a number of samples that is, up to a logarithmic factor, linear in the number of basis functions considered. We demonstrate these different sampling strategies numerically in both high-order and high-dimensional manufactured PC expansions. In addition, the quality of each sampling method is compared in the identification of solutions to two differential equations, one with a high-dimensional random input and the other with a high-order PC expansion. In all observed cases the coherence-optimal sampling leads to similar or considerably improved accuracy over the other considered sampling distributions.
TL;DR: In this article, a method for including coated structures and prescribed material interface properties into the minimum compliance topology optimization problem is presented, which is applicable to a broader range of interface problems.
Abstract: This paper presents a novel method for including coated structures and prescribed material interface properties into the minimum compliance topology optimization problem Several elements of the method are applicable to a broader range of interface problems The approach extends the standard SIMP method by including the normalized norm of the spatial gradient of the design field into the material interpolation function, enforcing coating material at interfaces by attributing particular properties The length scales of the base structure and the coating are separated by introducing a two-step filtering/projection approach The modeled coating thickness is derived analytically, and the coating is shown to be accurately controlled and applied in a highly uniform manner over the structure An alternative interpretation of the model is to perform single-material design for additive manufacturing Infill is assumed to be constituted of an isotropic porous microstructure satisfying the Hashin–Shtrikman bounds and is modeled using the homogenized material properties A range of numerical results illustrate the effectiveness of the approach
TL;DR: In this paper, a well-scaled projection-contraction algorithm is designed to simulate growth of multiple cracks in a natural way, allowing the crack tips to stop anywhere with no sensitivity to node configuration or cracking increments.
Abstract: The full response of a brittle structure containing multiple cracks to loading under the servo control is of vital importance in the evaluation of properties of the structure. During crack growth, the fracture toughness condition at all the crack tips as well as the equilibrium condition should be obeyed, leading to a nonlinear complementarity problem (NCP). The vector-valued function in the NCP depends implicitly on the cracking increments which in turn determine the stress field. The stress field can be obtained through solving a mixed variational problem. Since the degrees of freedom in the discrete variational problem vary with cracking not only in magnitude but also in number, the Jacobian matrix of the NCP is hard to compute and, it cannot be expected to be solved by the Newton methods that involve the calculation of Jacobian matrices. Therefore, a well-scaled projection-contraction algorithm is designed. The proposed procedure is able to simulate growth of multiple cracks in a natural way, allowing the crack tips to stop anywhere with no sensitivity to node configuration or cracking increments. Through the analysis of some examples that have been widely tested, many interesting and profound phenomena are found which have never been revealed.
TL;DR: In this article, the authors present an approach to modeling the morphological and crystallographic reorientation associated with the formation and thickening of a twin lamella within a crystal plasticity finite element (CPFE) framework.
Abstract: Deformation twinning is a subgrain mechanism that strongly influences the mechanical response and microstructural evolution of metals especially those with low symmetry crystal structure. In this work, we present an approach to modeling the morphological and crystallographic reorientation associated with the formation and thickening of a twin lamella within a crystal plasticity finite element (CPFE) framework. The CPFE model is modified for the first time to include the shear transformation strain associated with deformation twinning. Using this model, we study the stress–strain fields and relative activities of the active deformation modes before and after the formation of a twin and during thickening within the twin, and in the parent grain close to the twin and away from the twin boundaries. These calculations are carried out in cast uranium (U), which has an orthorhombic crystal structure and twins predominantly on the { 130 } 3 1 0 > systems under ambient conditions. The results show that the resolved shear stresses on a given twin system on the twin–parent grain interface and in the parent are highly inhomogeneous. We use the calculated mechanical fields to determine whether the twin evolution occurs via thickening of the existing twin lamella or formation of a second twin lamella. The analysis suggests that the driving force for thickening the existing twin lamella is low and that formation of multiple twin lamellae is energetically more favorable. The overall modeling framework and insight into why twins in U tend to be thin are described and discussed in this paper.
TL;DR: In this article, the performance of variational multiscale models (VMS) in the large eddy simulation (LES) of turbulent flows is studied, and the results show the tremendous potential of VMS for the numerical simulation of turbulence.
Abstract: In this work we study the performance of some variational multiscale models (VMS) in the large eddy simulation (LES) of turbulent flows. We consider VMS models obtained by different subgrid scale approximations which include either static or dynamic subscales, linear or nonlinear multiscale splitting, and different choices of the subscale space. After a brief review of these models, we discuss some implementation aspects particularly relevant to the simulation of turbulent flows, namely the use of a skew symmetric form of the convective term and the computation of projections when orthogonal subscales are used. We analyze the energy conservation (and numerical dissipation) of the alternative VMS formulations, which is numerically evaluated. In the numerical study, we have considered three well known problems: the decay of homogeneous isotropic turbulence, the Taylor–Green vortex problem and the turbulent flow in a channel. We compare the results obtained using different VMS models, paying special attention to the effect of using orthogonal subscale spaces. The VMS results are also compared against classical LES scheme based on filtering and the dynamic Smagorinsky closure. Altogether, our results show the tremendous potential of VMS for the numerical simulation of turbulence. Further, we study the sensitivity of VMS to the algorithmic constants and analyze the behavior in the small time step limit. We have also carried out a computational cost comparison of the different formulations. Out of these experiments, we can state that the numerical results obtained with the different VMS formulations (as far as they converge) are quite similar. However, some choices are prone to instabilities and the results obtained in terms of computational cost are certainly different. The dynamic orthogonal subscales model turns out to be best in terms of efficiency and robustness.
TL;DR: In this paper, an energy-based crack tracking strategy, originally used in the framework of the XFEM, is modified and implemented into the Strong Discontinuity Embedded Approach (SDA) model.
Abstract: The Strong Discontinuity embedded Approach (SDA) has proved to be a robust numerical method for simulating fracture of quasi-brittle materials such as glass and concrete. Three different numerical formulations are used in the SDA: (i) Statically Optimal Symmetric (SOS), (ii) Kinematically Optimal Symmetric (KOS), (iii) Statically and Kinematically Optimal Nonsymmetric (SKON). While the SOS formulation (standard version) of the SDA is simple for coding and provides good numerical stability (considering standard Galerkin method), several researchers have pointed out that this formulation encounters serious stress-locking. In this paper, an SOS formulation of the SDA is presented, considering elements with linear and quadratic interpolation for the displacement field. The performance of the proposed model of crack simulation is investigated by re-analysis of a pulling test, a three-point bending test, and an L-shaped panel with prescribed crack paths. The obtained results show that elements with linear interpolation encounter significant stress-locking, whereas elements with quadratic interpolation give good results without locking. Based on the eight-node quadrilateral element, which showed the best performance in the aforementioned re-analysis of the tests, an energy-based crack-tracking strategy, originally used in the framework of the XFEM, is modified and implemented into the SDA model. Several numerical benchmark tests, including an L-shaped panel test, a tension-shear test, a notched panel test, four-point bending tests with single and double notches, and a pull-out test are performed, illustrating the good performance with respected to the SDA model and the robustness of the proposed crack-tracking strategy.