Simulating non‐holonomic constraints within the LCP‐based simulation framework

In this paper, we will extend the linear complementarity problem‐based rigid‐body simulation framework with non‐holonomic constraints. We consider three different types of such, namely equality, inequality and contact constraints. We show how non‐holonomic equality and inequality constraints can be incorporated directly, and derive formalism for how the non‐holonomic contact constraints can be modelled as a combination of non‐holonomic equality constraints and ordinary contacts constraints. For each of these three we are able to guarantee solvability, when using Lemke's algorithm. A number of examples are included to demonstrate the non‐holonomic constraints. Copyright © 2006 John Wiley & Sons, Ltd     

Enrichment based multiscale modeling for thermo‐stress analysis of heterogeneous material

This paper details a novel new multiscale technique for modeling heterogeneous materials undergoing substantial thermal stresses. The technique is based on an enriched partition of unity approach that incorporates the thermal effects occurring on the microstructure into the global model. We demonstrate the effectiveness of this technique by implementing it into both the standard finite element method and the octree partition of unity method (OctPUM). The results demonstrate that the technique has uniquely improved accuracy over the homogenization method conditional to the method into which it is implemented in. The multiscale technique, when implemented into either the standard finite element method or OctPUM, increases the accuracy of the strain energy calculation. When the multiscale technique is implemented into OctPUM, it also is able to capture the unique stress fields on the microstructure of the model. Published 2012. This article is a US Government work and is in the public domain in the USA.     

Numerical finite element formulation of the Schapery non‐linear viscoelastic material model

This study presents a numerical integration method for the non‐linear viscoelastic behaviour of isotropic materials and structures. The Schapery's three‐dimensional (3D) non‐linear viscoelastic material model is integrated within a displacement‐based finite element (FE) environment. The deviatoric and volumetric responses are decoupled and the strain vector is decomposed into instantaneous and hereditary parts. The hereditary strains are updated at the end of each time increment using a recursive formulation. The constitutive equations are expressed in an incremental form for each time step, assuming a constant incremental strain rate. A new iterative procedure with predictor–corrector type steps is combined with the recursive integration method. A general polynomial form for the parameters of the non‐linear Schapery model is proposed. The consistent algorithmic tangent stiffness matrix is realized and used to enhance convergence and help achieve a correct convergent state. Verifications of the proposed numerical formulation are performed and compared with a previous work using experimental data for a glassy amorphous polymer PMMA. Copyright © 2003 John Wiley & Sons, Ltd.     

Umbrella spherical integration: a stable meshless method for non‐linear solids

A stable meshless method for studying the finite deformation of non‐linear three‐dimensional (3D) solids is presented. The method is based on a variational framework with the necessary integrals evaluated through nodal integration. The method is truly meshless, requiring no 3D meshing or tessellation of any form. A local least‐squares approximation about each node is used to obtain necessary deformation gradients. The use of a local field approximation makes automatic grid refinement and the application of boundary conditions straightforward. Stabilization is achieved through the use of special ‘umbrella’ shape functions that have discontinuous derivatives at the nodes. Novel efficient algorithms for constructing the nodal stars and computing the nodal volumes are presented. The method is applied to four test problems: uniaxial tension, simple shear and bending of a bar, and cylindrical indentation. Convergence studies at infinitesimal strain show that the method is well‐behaved and converges with the number of nodes for both uniform and non‐uniform grids. Typical of meshless methods employing nodal integration, the total energy can be underestimated due to the approximate integration. At finite deformation the method reproduces known exact solutions. The bending example demonstrates an interesting example of torsional buckling resulting from the bending. Copyright © 2006 John Wiley & Sons, Ltd.     

An approach to multi‐start clustering for global optimization with non‐linear constraints

This paper describes a multi‐start with clustering strategy for use on constrained optimization problems. It is based on the characteristics of non‐linear constrained global optimization problems and extends a strategy previously tested on unconstrained problems. Earlier studies of multi‐start with clustering found in the literature have focused on unconstrained problems with little attention to non‐linear constrained problems. In this study, variations of multi‐start with clustering are considered including a simulated annealing or random search procedure for sampling the design domain and a quadratic programming (QP) sub‐problem used in cluster formation. The strategies are evaluated by solving 18 non‐linear mathematical problems and six engineering design problems. Numerical results show that the solution of a one‐step QP sub‐problem helps predict possible regions of attraction of local minima and can enhance robustness and effectiveness in identifying local minima without sacrificing efficiency. In comparison to other multi‐start techniques found in the literature, the strategies of this study can be attractive in terms of the number of local searches performed, the number of minima found, whether the global minimum is located, and the number of the function evaluations required. Copyright © 2002 John Wiley & Sons, Ltd.     

Robust multiscale algorithms for gradient‐based motion estimation

Gradient‐based techniques represent a very popular class of approaches to estimate motions. A robust multiscale algorithm of hierarchical estimation for gradient‐based motion estimation is proposed in this article using a combination of robust statistical method and multiscale technique. In such a multiscale approach of hierarchical estimation, motion at each level of the pyramid is estimated using different gradient filters. The iterative multiscale estimation begins by using five‐tap central filter, and it is switched to nine‐tap Timoner filter after a few iterations. In addition, robust M‐estimators are applied at each level of the pyramid to overcome the problem of the outliers caused by illumination variations and motion discontinuities in motion estimation. Experimental simulations show that the new algorithm not only provides an improvement in estimator accuracy, but also achieves computational speedups. © 2008 Wiley Periodicals, Inc. Int J Imaging Syst Technol, 17, 333–340, 2007     

Coarse‐graining of multiscale crack propagation

A method for coarse graining of microcrack growth to the macroscale through the multiscale aggregating discontinuity (MAD) method is further developed. Three new features are: (1) methods for treating nucleating cracks, (2) the linking of the micro unit cell with the macroelement by the hourglass mode, and (3) methods for recovering macrocracks with variable crack opening. Unlike in the original MAD method, ellipticity is not retained at the macroscale in the bulk material, but we show that the element stiffness of the bulk material is positive definite. Several examples with comparisons with direct numerical simulations are given to demonstrate the effectiveness of the method. Copyright © 2009 John Wiley & Sons, Ltd.     

Variational multiscale enrichment for modeling coupled mechano‐diffusion problems

In this paper, a new multiscale–multiphysics computational methodology is devised for the analysis of coupled diffusion–deformation problems. The proposed methodology is based on the variational multiscale principles. The basic premise of the approach is accurate fine‐scale representation at a small subdomain where it is known a priori that important physical phenomena are likely to occur. The response within the remainder of the problem domain is idealized on the basis of coarse‐scale representation. We apply this idea to evaluate a coupled mechano‐diffusion problem that idealizes the response of titanium structures subjected to a thermo–chemo–mechanical environment. The proposed methodology is used to devise a multiscale model in which the transport of oxygen into titanium is modeled as a diffusion process, whereas the mechanical response is idealized using concentration‐dependent elasticity equations. A coupled solution strategy based on operator split is formulated to evaluate the coupled multiphysics and multiscale problem. Numerical experiments are conducted to assess the accuracy and computational performance of the proposed methodology. Numerical simulations indicate that the variational multiscale enrichment has reasonable accuracy and is computationally efficient in modeling the coupled mechano‐diffusion response. Copyright © 2011 John Wiley & Sons, Ltd.     

A nonlocal multiscale discrete‐continuum model for predicting mechanical behavior of granular materials

A three‐dimensional nonlocal multiscale discrete‐continuum model has been developed for modeling mechanical behavior of granular materials. In the proposed multiscale scheme, we establish an information‐passing coupling between the discrete element method, which explicitly replicates granular motion of individual particles, and a finite element continuum model, which captures nonlocal overall responses of the granular assemblies. The resulting multiscale discrete‐continuum coupling method retains the simplicity and efficiency of a continuum‐based finite element model, while circumventing mesh pathology in the post‐bifurcation regime by means of staggered nonlocal operator. We demonstrate that the multiscale coupling scheme is able to capture the plastic dilatancy and pressure‐sensitive frictional responses commonly observed inside dilatant shear bands, without employing a phenomenological plasticity model at a macroscopic level. In addition, internal variables, such as plastic dilatancy and plastic flow direction, are now inferred directly from granular physics, without introducing unnecessary empirical relations and phenomenology. The simple shear and the biaxial compression tests are used to analyze the onset and evolution of shear bands in granular materials and sensitivity to mesh density. The robustness and the accuracy of the proposed multiscale model are verified in comparisons with single‐scale benchmark discrete element method simulations. Copyright © 2015 John Wiley & Sons, Ltd.     

Integrating a micromechanical model for multiscale analyses

The Chang‐Hicher micromechanical model based on a static hypothesis, not unlike other models developed separately at around the same epoch, has proved its efficiency in predicting soil behaviour. For solving boundary value problems, the model has now integrated stress‐strain relationships by considering both the micro and macro levels. The first step was to solve the linearized mixed control constraints by the introduction of a predictor–corrector scheme and then to implement the micro–macro relationships through an iterative procedure. Two return mapping schemes, consisting of the closest‐point projection method and the cutting plane algorithm, were subsequently integrated into the interparticle force‐displacement relations. Both algorithms have proved to be efficient in studies devoted to elementary tests and boundary value problems. Closest‐point projection method compared with cutting plane algorithm, however, has the advantage of being more intensive cost efficient and just as accurate in the computational task of integrating the local laws into the micromechanical model. The results obtained demonstrate that the proposed linearized method is capable of performing loadings under stress and strain control. Finally, by applying a finite element analysis with a biaxial test and a square footing, it can be recognized that the Chang‐Hicher micromechanical model performs efficiently in multiscale modelling.     

Ballistic performance study of fabric armor based on numerical simulations with multiscale material model

Based on a multiscale model for fabric materials, dynamic simulations of the fabric ballistic performance were implemented. Through parameter research, it was found that the ballistic performance and mechanical behavior of the fabric materials are determined by a combination of factors and conditions rather than by the material properties alone. The material mechanical properties reflect the inherent strength of the fabric; the fabric weaving structure, boundary conditions, material orientation, and projectile shape also play important roles and have a significant influence on the ballistic performance of the fabric. The multiscale material model incorporates not only the membrane‐like properties of the fabric but also the underlying weaving structure, yarn interaction, and yarn composition. The simulations results show good agreement with the experimental data. Various physical phenomena can be observed in the simulations, such as yarn decrimping, material anisotropy, and two types of damage modes. Copyright © 2011 John Wiley & Sons, Ltd.     

A multilevel projection‐based model order reduction framework for nonlinear dynamic multiscale problems in structural and solid mechanics

A reduction/hyper reduction framework is presented for dramatically accelerating the solution of nonlinear dynamic multiscale problems in structural and solid mechanics. At each scale, the dimensionality of the governing equations is reduced using the method of snapshots for proper orthogonal decomposition, and computational efficiency is achieved for the evaluation of the nonlinear reduced‐order terms using a carefully designed configuration of the energy conserving sampling and weighting method. Periodic boundary conditions at the microscales are treated as linear multipoint constraints and reduced via projection onto the span of a basis formed from the singular value decomposition of Lagrange multiplier snapshots. Most importantly, information is efficiently transmitted between the scales without incurring high‐dimensional operations. In this proposed proper orthogonal decomposition–energy conserving sampling and weighting nonlinear model reduction framework, training is performed in two steps. First, a microscale hyper reduced‐order model is constructed in situ, or using a mesh coarsening strategy, in order to achieve significant speedups even in non‐parametric settings. Next, a classical offline–online training approach is performed to build a parametric hyper reduced‐order macroscale model, which completes the construction of a fully hyper reduced‐order parametric multiscale model capable of fast and accurate multiscale simulations. A notable feature of this computational framework is the minimization, at the macroscale level, of the cost of the offline training using the in situ or coarsely trained hyper reduced‐order microscale model to accelerate snapshot acquisition. The effectiveness of the proposed hyper reduction framework at accelerating the solution of nonlinear dynamic multiscale problems is demonstrated for two problems in structural and solid mechanics. Speedup factors as high as five orders of magnitude are shown to be achievable. Copyright © 2017 John Wiley & Sons, Ltd.     

Two‐level multiscale enrichment methodology for modeling of heterogeneous plates

A new two‐level multiscale enrichment methodology for analysis of heterogeneous plates is presented. The enrichments are applied in the displacement and strain levels: the displacement field of a Reissner–Mindlin plate is enriched using the multiscale enrichment functions based on the partition of unity principle; the strain field is enriched using the mathematical homogenization theory. The proposed methodology is implemented for linear and failure analysis of brittle heterogeneous plates. The eigendeformation‐based model reduction approach is employed to efficiently evaluate the non‐linear processes in case of failure. The capabilities of the proposed methodology are verified against direct three‐dimensional finite element models with full resolution of the microstructure. Copyright © 2009 John Wiley & Sons, Ltd.     

Multilayered grouping parallel algorithm for multiple‐level multiscale analyses

Multiscale analysis technique became a successful remedy to complicated problems in which nonlinear behavior is linked with microscopic damage mechanisms. For efficient parallel multiscale analyses, hierarchical grouping algorithms (e.g., the two‐level ‘coarse‐grained’ method) have been suggested and proved superior over a simple parallelization. Here, we expanded the two‐level algorithm to give rise to a multilayered grouping parallel algorithm suitable for large‐scale multiple‐level multiscale analyses. With practical large‐scale applications, we demonstrated the superior performance of multilayered grouping over the coarse‐grained method. Notably, we show that the unique data transfer rates of the symmetric multiprocessor cluster system can lead to the seemingly ‘super‐linear speedup’ and that there appears to exist the optimal number of subgroups of three‐tiered multiscale analysis. Copyright © 2014 John Wiley & Sons, Ltd.     

Optimum design of stochastically excited non‐linear dynamic systems without geometric constraints

This paper presents an efficient procedure for min–max dynamic response optimization of stochastically excited non‐linear systems with multiple time‐delayed inputs. This procedure employs a stochastic linearization technique to overcome system non‐linearity and an auto‐covariance analysis technique to represent the original stochastic mechanical model in a suitable form for optimization. Special attention is given to the sensitivity analysis, due to the complex nature of the problem. Therefore, exact expressions are obtained in a simple form for the evaluation of the required gradients, which greatly improve the stability and efficiency of the optimization algorithm. The numerical results and performance are presented by means of solving two min–max dynamic response optimization problems. Copyright © 2002 John Wiley & Sons, Ltd.     

Minimum weight design of non‐linear elastic structures with multimodal buckling constraints

It well known that multimodal instability is an event particularly relevant in structural optimization. Here, in the context of non‐linear stability theory, an exact method is developed for minimum weight design of elastic structures with multimodal buckling constraints. Given an initial design, the method generates a sequence of improved designs by determining a sequence of critical equilibrium points related to decreasing values of the structural weight. Multimodal buckling constraints are imposed without repeatedly solving an eigenvalue problem, and the difficulties related to the non‐differentiability in the common sense of state variables in multimodal critical states, are overcome by means of the Lagrange multiplier method. Further constraints impose that only the first critical equilibrium states (local maxima or bifurcation points) on the initial equilibrium path of the actual designs are taken into account. Copyright © 2002 John Wiley & Sons, Ltd.     

Reliability‐based design optimization with equality constraints

Equality constraints have been well studied and widely used in deterministic optimization, but they have rarely been addressed in reliability‐based design optimization (RBDO). The inclusion of an equality constraint in RBDO results in dependency among random variables. Theoretically, one random variable can be substituted in terms of remaining random variables given an equality constraint; and the equality constraint can then be eliminated. However, in practice, eliminating an equality constraint may be difficult or impossible because of complexities such as coupling, recursion, high dimensionality, non‐linearity, implicit formats, and high computational costs. The objective of this work is to develop a methodology to model equality constraints and a numerical procedure to solve a RBDO problem with equality constraints. Equality constraints are classified into demand‐based type and physics‐based type. A sequential optimization and reliability analysis strategy is used to solve RBDO with physics‐based equality constraints. The first‐order reliability method is employed for reliability analysis. The proposed method is illustrated by a mathematical example and a two‐member frame design problem. Copyright © 2007 John Wiley & Sons, Ltd.     

A staggered nonlocal multiscale model for a heterogeneous medium

A staggered nonlocal multiscale model for a heterogeneous medium is developed and validated. The model is termed as staggered nonlocal in the sense that it employs current information for the point under consideration and past information from its local neighborhood. For heterogeneous materials, the concept of phase nonlocality is introduced by which nonlocal phase eigenstrains are computed using different nonlocal phase kernels. The staggered nonlocal multiscale model has been found to be insensitive to finite element mesh size and load increment size. Furthermore, the computational overhead in dealing with nonlocal information is mitigated by superior convergence of the Newton method. Copyright © 2012 John Wiley & Sons, Ltd.     

Computational schemes for non‐linear elasto‐dynamics

This paper deals with the development of computational schemes for the dynamic analysis of non‐linear elastic systems. The focus of the investigation is on the derivation of unconditionally stable time‐integration schemes presenting high‐frequency numerical dissipation for these types of problem. At first, schemes based on Galerkin and time‐discontinuous Galerkin approximations applied to the equations of motion written in the symmetric hyperbolic form are proposed. Though useful, these schemes require casting the equations of motion in the symmetric hyperbolic form, which is not always possible. Furthermore, this approaches to unacceptably high computational costs. Next, unconditionally stable schemes are proposed that do not rely on the symmetric hyperbolic form. Both energy‐preserving and energy‐decaying schemes are derived. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed schemes. Copyright © 1999 John Wiley & Sons, Ltd.     

Boundary value problems defined on stochastic self‐similar multiscale geometries

A method for solving elasticity problems defined on composite bodies with a stochastic multiscale microstructure is presented. It is considered that the composite is made from two types of materials with different elastic moduli. One of these is taken as the matrix, while the other forms the inclusions. The inclusions form a stochastic fractal with a finite, but potentially large, number of scales and are randomly distributed within the matrix. The method presented here leads to the statistics of the solution, i.e. the mean and the variance of the stress and displacement fields. It is based on the stochastic finite element method (spectral approach, second‐order technique) and on scaling properties of the spatial distribution of inclusions over the problem domain. This scaling allows for a simple formulation of the multiscale problem and leads to significant computation cost savings, especially when the fractal has a large number of relevant scales. Several examples are presented and used to verify the proposed method against computationally intensive classical finite element models in which the mesh is refined down to the scale of the finest inclusions. Copyright © 2007 John Wiley & Sons, Ltd.