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Detecting and characterising shocks is challenging because they do not occur in isolation but are instead superimposed onto underlying vehicle vibrations which themselves are a result of the interaction with uneven road surfaces. Consequently, shocks are buried within vehicle vibration response measurements (usually acceleration). This paper presents the development and validation of an automated algorithm to detect shocks from vertical acceleration signals measured from road vehicles. To avoid inherent difficulties with experimentation, this initial paper is confined to numerical simulation whereby the response of two typical quarter-car truck models (one with air ride and the other with steel suspension) when travelling on artificially-generated random road elevation profiles laced with Hanning-shaped surface aberrations of known amplitude, lengths and location along the elevation profile. The shock detection algorithm was developed as a dual mode classifier to accommodate the two natural frequencies of each 2DoF quarter car models. Detection was implemented by first passing the vibration response signal through a band-pass filter around the two resonant modes in turn then calculating the filtered signals' instantaneous frequency (envelope) by means of the Hilbert transform. Shock detection was based on the local peak-to-mean ratio (LPTMR) of the instantaneous magnitude where 'local' was defined by the duration of the filtered signal's impulse response function. Sensitivity analysis and validation were undertaken on artificially-generated roads of varying roughness onto which aberrations of known shape were superimposed. The effectiveness and limitations (detection threshold) of the algorithm were evaluated by creating a range of aberrations with a broad range of lengths (effective frequencies) and diminishing amplitudes. Results show that shocks of 'significant' magnitudes are always detected with no detection of false positives. As the road roughness increases relative to the aberration amplitudes, the resulting shocks become increasingly drowned-out by the vibration response due to the underlying road roughness, especially for the quarter-car model with steel suspension. The main conclusion is that the signal analysis approach taken is ultimately effective and needs to be further validated using experimental response data. [ABSTRACT FROM AUTHOR]

The challenges of classic rate-independent crystal plasticity models regarding the determination of active slip systems and computation of solutions due to linear dependent slip systems are still a field of active research. In that regard, interior-point methods are promising approaches because a logarithmic barrier term naturally avoids the combinatorial active set search, and these methods are applicable even when the gradient of inequality constraints is linearly dependent. In this paper, a new interior-point algorithm suitable for rate-independent crystal plasticity at finite strains is developed. For that purpose, the postulate of maximum dissipation is recast as a constrained optimization problem whose optimality conditions are derived by utilizing interior-point approaches. This optimization problem is discretized in time by means of an exponential time integration of the flow rule, and the time-discrete equations are solved by a Newton scheme. The focus of this paper is the efficiency and robustness of the implementation of this approach. In particular, the effect of different line search procedures, favorable residual forms, preconditioning methods, barrier parameter updates and the consistent algorithmic tangent modulus are discussed and further elaborated. In addition, aspects regarding uniqueness and well-posedness of the algorithm are analyzed in a geometrically linearized setting. Efficiency and robustness of the developed algorithm are shown by numerical examples of an FCC crystal including finite element simulations without stabilizing hardening or rate-dependent effects. Furthermore, the results are compared to the well-established augmented Lagrangian formulation. [ABSTRACT FROM AUTHOR]

• An analytical accuracy framework is developed for the second-order-type time integration methods. • Three types of second-order-type linear multistep methods are revealed with second-order convergence and A-stability. • A single-step methods family is revealed and its displacement, velocity, and acceleration can achieve the same order of accuracy simultaneously. • A family of accuracy-improved single-step methods with algorithmic equivalence to the second-order-type linear multistep methods is disclosed. • An algorithm optimization strategy is proposed, which leads to an optimal single-step method with a single parameter to control the accuracy and numerical dissipation. This paper presents a comprehensive study of the second-order-type time integration methods family encompassing the linear multistep (LMS) methods, the single-step methods, and the LMS equivalent single-step methods for structural dynamics. An analytical accuracy framework for algorithm design, convergence, and optimization is developed, where the obstacles encountered in algorithms construction, convergence accuracy, algorithms equivalence, accuracy measurement, and accuracy optimization in traditional accuracy frameworks are well addressed. Rigorous accuracy analysis successively to the second-order-type LMS (LMS2) methods, the single-step methods, and the LMS2 equivalent single-step methods aims to reveal optimal algorithms with desirable numerical properties. First, independent implicit algorithms including the velocity type and the acceleration type LMS2 methods except the displacement type ones are newly revealed in the LMS2 methods family with second-order convergence and A-stability, and three error constants that can play essential roles in the accuracy measurement and algorithm optimization are developed. Second, a single-step methods family where the displacement, velocity, and acceleration can achieve the same order of accuracy simultaneously is revealed in a generalized 16-parameters methods family, which contains numerous conventional single-step single-solve methods as subfamily methods. It is found that the unit consistent and non-consistent representations show a significant difference in the local truncation error, stability, and convergence performances. Third, a family of accuracy-improved single-step methods with algorithmic equivalence, which is stricter than the traditional spectral equivalence, to the LMS2 methods is revealed. It is clarified that the difference between the methods with equivalence can significantly influence their convergence accuracy, and the error constants concept of the LMS2 methods can be extended to the single-step method with algorithmic equivalence. Last, an algorithm optimization strategy is developed based on the error constants and the ultimate spectral radius, and an optimal single-step method with a single parameter to control the accuracy and numerical dissipation is revealed analytically without empirical assumptions. The optimal method is second-order accurate, unconditionally stable, overshooting acceptable, and equivalent to an optimal route between the Houbolt method and the trapezoidal rule, and its numerical performance is verified by the spectral accuracy and two numerical examples. [ABSTRACT FROM AUTHOR]

The recently introduced Artificial Hummingbird Algorithm (AHA) exhibits competitive performance in developing optimization concerns. However, AHA has an imbalance between exploration and utilization abilities, often prematurely converging with low precision. Therefore, in this paper, a multi-strategy boosted hybrid artificial hummingbird algorithm called LCAHA combined with sinusoidal chaotic map strategy, Lévy flight, cross, and update foraging strategy is proposed. Firstly, LCAHA is initialized by the sinusoidal chaotic map strategy to obtain a population with better ergodicity. Secondly, introducing the Lévy flight can boost the diversity of the population, control premature convergence and boost the stability of the population. Then, two new strategies, cross foraging and update foraging, are introduced. The introduction of new foraging strategies further enhances the exploration and developmental capabilities. These three strategies work together to improve the overall performance of the AHA. Finally, the performance of the LCAHA is examined on 23 classical test suites, the CEC2017, CEC2019, and CEC2020 test suites, and six engineering optimization cases. The numerical experimental results show that LCAHA provides very promising numerical results in solving challenging optimization problems. The algorithm is applied to two spacecraft trajectory optimization cases. The experimental results demonstrate the applicability and potential of the LCAHA in solving practical applications. [ABSTRACT FROM AUTHOR]

Many applications in additive manufacturing involve the dynamic deposition of powders and the in-flight heating of such material by a laser. In order to characterize such systems, the Discrete Element Method (DEM) is employed. Specifically, this paper focuses on the initial stages of this process by developing a modular discrete-element type multiphysics simulation method for the particle dynamics and heating by a laser and detailed thermal behavior. The objective is to provide researchers with a framework to construct computational tools for this growing industry. In order to achieve this, from a simulation standpoint, the overall particle-mixture system is constructed by coupling submodels for each primary physical process (dynamics and heating) together. An iterative staggering scheme is developed whereby, within every time step, each individual particle is solved “freezing” the state of the remaining multiparticle system. The state of the particle is then updated and the algorithm moves to the next particle in the system and the process is repeated. The overall process sweeps through the entire system repeatedly until convergence is achieved in an appropriate norm. As the system evolves, an error estimate dictates the time-step size that is needed to induce convergence to below an appropriate error level. Thus, the process can be considered as an implicit time-stepping scheme, which is combined with an (internal) iterative staggering process. In order to control rates of convergence within a time-step, the algorithm adjusts the time-step size. If the iterative process does not converge within a desired number of iterations, below an error tolerance, the time-step is reduced. The degree of time-step reduction is determined by utilizing an estimate of the spectral radius of the coupled system. Since the construction of model and solution process is modular, one can easily replace physical submodels with other choices, making it easy to numerically experiment with a variety of models. Qualitative and quantitative analyses are provided, as well as three-dimensional numerical examples. This paper addresses the first of two overall “macrostages” of a complex manufacturing process with DEM. The first stage, addressed in this paper, concentrates on dry-powder materials, which are of a discrete particulate character, using DEM to characterize the deposition and in-flight particle heating. Other papers of the author (Zohdi, 2010, 2013, 2014, 2015 [81,82], 2008) have addressed stage two, namely phase transformations, curing (cooling) and stress analysis, using more appropriate continuum and hybrid DEM-continuum approaches. [ABSTRACT FROM AUTHOR]

Meshfree approximation schemes possess a high potential in computer aided engineering due to their large flexibility. Especially the tremendous progress in processor technology within recent years relativizes the increase in computation time due to the inherent search algorithm. Nevertheless meshfree approximation schemes are still faced with some challenges, like imposition of Dirichlet boundary conditions, robustness of the algorithm and accuracy. The recent developed Optimal Transportation Meshfree (OTM) method seemed to overcome most of these problems. In this paper the OTM solution scheme is combined with a standard search algorithm in order to allow a simple and flexible computation. However this scheme is not stable for some examples of application. Hence an investigation is conducted which shows that the reason for this instability is due to underintegration. Based on this investigation a remedy to stabilize the algorithm is suggested which is based on well known concepts to control the hourglass effects in the Finite Element Method. In contrast to the original publication, the OTM algorithm is derived here from the principle of virtual work. Local maximum entropy shape functions are used which possess a weak Kronecker- δ property. This enables a direct imposition of Dirichlet boundary conditions if the boundary is convex. The limitations of this basis function are also addressed in this paper. Additionally, the search algorithm presented fulfills basic topological requirements. Several examples are investigated demonstrating the improved behavior of the stabilized OTM algorithm. [ABSTRACT FROM AUTHOR]

This paper extends the non-uniform rational basis spline (NURBS) plasticity framework of Coombs et al. (2016) to include isotropic hardening of the yield surfaces. The approach allows any smooth isotropic yield envelope to be represented by a NURBS surface. The key extension provided by this paper is that the yield surface can expand or contract through the movement of control points linked to the level of inelastic straining experienced by the material. The model is integrated using a fully implicit backward Euler algorithm that constrains the return path to the yield surface and allows the derivation of the algorithmic consistent tangent to ensure optimum convergence of the global equilibrium equations. This provides a powerful framework for elasto-plastic constitutive models where, unlike the majority of models presented in the literature, the underlying numerical algorithm (and implemented code) remains unchanged for different yield surfaces. The performance of the algorithm is demonstrated, and validated, using both material point and boundary values simulations including plane stress, plane strain and three-dimensional examples for different yield criteria. [ABSTRACT FROM AUTHOR]

In this paper, we consider numerical approximations for solving the anisotropic Cahn–Hilliard model. We combine the Scalar Auxiliary Variable (SAV) approach with the stabilization technique to arrive at a novel Stabilized-SAV approach, where three linear stabilization terms, which are shown to be crucial to remove the oscillations caused by the anisotropic coefficient, are added to enhance the stability while keeping the required accuracy. The schemes are very easy-to-implement and fast in the sense that all nonlinear terms are treated in a semi-explicit way, and one only needs to solve three decoupled linear equations with constant coefficients at each time step. We further prove the unconditional energy stabilities rigorously and present numerous 2D and 3D numerical simulations to demonstrate the stability and accuracy. • We propose a novel Stabilized-SAV approach for solving the anisotropic Cahn–Hilliard model. • The scheme is linear, second-order accurate, provably unconditionally energy stable, and non-iterative. • Three crucial linear stabilization terms are added to remove oscillations caused by anisotropy. • One only need to solve three decoupled linear equations at each time step. • We further prove the unconditional energy stabilities rigorously and present numerous simulations. [ABSTRACT FROM AUTHOR]

This paper presents a new Smooth Particle Hydrodynamics (SPH) computational framework for large strain explicit solid dynamics. A mixed-based set of Total Lagrangian conservation laws (Bonet et al., 2015; Gil et al., 2016) is presented in terms of the linear momentum and an extended set of geometric strain measures, comprised of the deformation gradient, its co-factor and the Jacobian. Taking advantage of this representation, the main aim of this paper is the adaptation of the very efficient Jameson–Schmidt–Turkel (JST) algorithm (Jameson et al., 1981) extensively used in computational fluid dynamics, to a SPH based discretisation of the mixed-based set of conservation laws, with three key distinct novelties. First, a conservative JST-based SPH computational framework is presented with emphasis in nearly incompressible materials. Second, the suppression of numerical instabilities associated with the non-physical zero-energy modes is addressed through a well-established stabilisation procedure. Third, the use of a discrete angular momentum projection algorithm is presented in conjunction with a monolithic Total Variation Diminishing Runge–Kutta time integrator in order to guarantee the global conservation of angular momentum. For completeness, exact enforcement of essential boundary conditions is incorporated through the use of a Lagrange multiplier projection technique. A series of challenging numerical examples (e.g. in the near incompressibility regime) are examined in order to assess the robustness and accuracy of the proposed algorithm. The obtained results are benchmarked against a wide spectrum of alternative numerical strategies. [ABSTRACT FROM AUTHOR]

Abstract Several theoretical contributions in order to establish a particle packing methodology are presented. In this respect, a generic formulation of a new method for packing particles based on a constructive advancing front method, which uses Monte Carlo techniques for the generation of particle dimensions, is also shown. The method can be used to obtain virtual dense packings of particles with several geometrical shapes. It employs continuous, discrete and empirical statistical distributions in order to generate the dimensions of particles. The packing algorithm is very flexible and allows alternatives for: 1- The direction of the advancing front (inwards or outwards), 2- The selection of the local advancing front, 3- The method for placing a mobile particle in contact with others and 4- The overlap checks. The algorithm also allows obtaining highly porous media when it is slightly modified. Practical applications of the formulations are presented in the end of this paper. [ABSTRACT FROM AUTHOR]

Abstract This paper presents a new approach for level set based shape optimization using trimmed hexahedral meshes that geometrically conform to shape and topology changes during the optimization process. The marching cubes algorithm is employed to generate trimmed hexahedral meshes at each optimization iteration by splitting a regular background hexahedral mesh with the zero-isosurface of a level set function. A simple method is proposed to handle nonconforming polyhedral elements at the interface due to the face ambiguity arising in the marching cubes algorithm. The motion of the zero-isosurface of a level set function is obtained from the solution of the Hamilton–Jacobi transport equation using a normal velocity field defined on the regular background hexahedral mesh. Numerical results show that the present approach provides a natural and effective tool to solve shape optimization problems relying on explicitly discretized meshes of the shapes throughout the optimization process. Highlights • Trimmed hexahedral meshes can be constructed by splitting a regular background hexahedral mesh with the zero-isosurface of a level set function. • The marching cubes algorithm is employed to determine the intersection configurations. • A simple method is proposed to handle nonconforming polyhedral elements at the interface due to the face ambiguity. • The structural domains during the optimization process can be clearly and explicitly discretized by trimmed hexahedral meshes. • The present approach provides a natural and effective tool to solve shape optimization problems relying on explicitly discretized meshes. [ABSTRACT FROM AUTHOR]

Abstract In previous work (Lee et al., 2016, 2017), Lee et al. introduced a new Smooth Particle Hydrodynamics (SPH) computational framework for large strain explicit solid dynamics with special emphasis on the treatment of near incompressibility. A first order system of hyperbolic equations was presented expressed in terms of the linear momentum and the minors of the deformation, namely the deformation gradient, its co-factor and its Jacobian. Taking advantage of this representation, the suppression of numerical deficiencies (e.g. spurious pressure, long term instability and/or consistency issues) was addressed through well-established stabilisation procedures. In Lee et al. (2016), the adaptation of the very efficient Jameson–Schmidt–Turkel algorithm was presented. Lee et al. (2017) introduced an adapted variationally consistent Streamline Upwind Petrov–Galerkin methodology. In this paper, we now introduce a third alternative stabilisation strategy, extremely competitive, and which does not require the selection of any user-defined artificial stabilisation parameter. Specifically, a characteristic-based Riemann solver in conjunction with a linear reconstruction procedure is used, with the aim to guarantee both consistency and conservation of the overall algorithm. We show that the proposed SPH formulation is very similar in nature to that of the upwind vertex centred Finite Volume Method presented in Aguirre et al. (2015). In order to extend the application range towards the incompressibility limit, an artificial compressibility algorithm is also developed. Finally, an extensive set of challenging numerical examples is analysed. The new SPH algorithm shows excellent behaviour in compressible, nearly incompressible and truly incompressible scenarios, yielding second order of convergence for velocities, deviatoric and volumetric components of the stress. [ABSTRACT FROM AUTHOR]

Abstract This paper is concentrated on the shape and topology optimization of 2D linear elastic problems using meshless element free Galerkin (EFG) method combined with the level set method (LSM). In this procedure, a level set function (LSF) is used for the implicit representation of the structural geometry, and thus ensuring appropriate essential topological changes to obtain optimized structures. Both the methods (EFG and LSM) use the same Cartesian grid points. Shape and topological sensitivities are evaluated by the EFG method and the structural geometry is updated through LSM. The algorithm is tested on five different standardized structural topology optimization problems for minimum compliance, subject to the application of load at single and multiple points. The results obtained with the proposed method suggest efficiency, convergence, accuracy and good agreement of the simulated results with the optimal topologies reported in the literature. Highlights • Structural optimization is performed using meshless procedure coupled with level set method. • Hamilton–Jacobi equation with topological derivative term is used to update the structure geometry. • Sensitivity analysis of the structural optimization problem is undertaken. • Simulation results are obtained for benchmark structural optimization problems. [ABSTRACT FROM AUTHOR]

Isogeometric analysis (IGA) is a novel computer aided engineering technique that addresses diverse problems in computational mechanics [ 1–4 ], all under the exact geometric representation. Apart from the exact geometric representation, the high continuity of IGA shape functions enhances the accuracy and robustness of the method. However, the price to pay is that the resulting matrices are denser, with increased bandwidth and overlapping which make the solution of large-scale problems more computationally intensive. As a result, effective solution techniques are still considered an open issue for further research. In this paper, an innovative family of solution schemes based on domain decomposition methods (DDM) is proposed, that significantly reduces the computational cost. Specifically, the solution of the global system is performed with the preconditioned conjugate gradient algorithm (PCG) whose preconditioning step is evaluated with a dual DDM where special care is taken to avoid the overlapped subdomains which are inherent in decomposed IGA formulation due to the increased continuity of the shape functions. [ABSTRACT FROM AUTHOR]

With the development of reliability technique, the safety assessment for the problem with epistemic uncertainty has attracted widespread attention. Evidence theory is a useful tool to deal with such uncertainty, and this paper aims to develop an efficient approach for the evidence theory-based reliability analysis and optimization design. By using mutually exclusive intervals to quantify the focal elements of evidence variable, a confidence range bounded by belief measure and plausibility measure is derived for system reliability assessment, by which the relatively conservative and radical optimization models can be respectively established. To decrease the huge computational burden in repetitive limit state function evaluations under the time-consuming implicit computational model, an explicit surrogate model is constructed by the Legendre polynomial chaos expansion in the support box. A Clenshaw–Curtis point-based collocation method with Smolyak algorithm is then developed to predict the unknown coefficients in surrogate model, where the collocation level can be flexibly selected according to the accuracy requirement. Compared with the traditional deterministic optimization model under nominal value assumption, the results in two numerical examples verify the effectiveness of proposed method in mathematical theory and engineering application. [ABSTRACT FROM AUTHOR]

The diameter of spot welds used in the automotive and aerospace industry is orders of magnitude smaller than the dimensions of the structural components. Automotive bodies typically have between three and five thousand spot welds which makes 3-D Direct Finite Element Analysis (DFEA) of this class problem not practical. To circumvent these limitations of the FEM, spot welds are idealized in industrial problems as elastic or rigid beams connecting nodes of the metal sheet meshes. However, parameters used in these spot weld models are often problem dependent. This paper presents a parallel Generalized Finite Element Method (GFEM) for the simulation of spot welds in large structures. The proposed GFEM can adopt structural-scale meshes that ignore the spot welds and thus can be generated much quicker than in a DFEA. The element size in the GFEM mesh is of the same order as those used in the industry for the class of problems considered here. Verification problems show that the proposed method can provide an accuracy comparable to a 3-D DFEA. Numerical experiments on a hat-stiffened panel with 168 spot welds show that the proposed GFEM scales much better than a DFEA on shared memory machines, in particular when nonlinear material behavior near the spot welds is considered. [ABSTRACT FROM AUTHOR]

This paper shows the recent work of the authors in the development of a time-domain FEM model for evaluation of the seal dynamics of a surface effect ship. The fluid solver developed for this purpose, uses a potential flow approach along with a stream-line integration of the free surface. The paper focuses on the free surface-structure algorithm that has been developed to allow the simulation of the complex and highly dynamic behaviour of the seals in the interface between the air cushion, and the water. The developed fluid–structure interaction solver is based, on one side, on an implicit iteration algorithm, communicating pressure forces and displacements of the seals at memory level and, on the other side, on an innovative wetting and drying scheme able to predict the water action on the seals. The stability of the iterative scheme is improved by means of relaxation, and the convergence is accelerated using Aitken’s method. Several validations against experimental results have been carried out to demonstrate the developed algorithm. [ABSTRACT FROM AUTHOR]

In order to maintain the quadratic convergence properties of Newton’s method in quasi-static nonlinear analysis of solid structures it is crucial to obtain accurate, algorithmically consistent tangent-stiffness matrices. A goal of the study described in this paper was to establish the suitability of an underexplored method for numerical computation of tangent-stiffness operators, referred to as “complex-step”, and compare the method with other techniques for numerical derivative calculation: automatic differentiation, forward finite difference, and central finite difference. The complex-step method was implemented in a massively parallel computational peridynamics code for the purpose of this comparison. The methods were compared through profiling of the code for accuracy, speed, efficiency, and parallel scalability. This research provides data that can serve as practical guide for code developers and analysts faced with choosing which method best suits the needs of their application code. Additionally, motivated by the reproducible research movement, all of the code, examples, and workflow to regenerate the data and figures in this paper are provided as open source. [ABSTRACT FROM AUTHOR]

In this paper, we propose an efficient contact algorithm to analyze the interaction of material particles with finite elements. The contact forces between material particles and finite elements are computed at the (background) grid nodes in material point method, and transferred to finite element nodes. A weighted average method is used to obtain the velocities at grid nodes contributed from finite element nodes, and a cut-off weight is applied to the contact condition to avoid an early contact occurring in a grid-based contact algorithm. A severe penetration of material particles into finite elements when the finite element mesh is coarser than the background grid can be prevented by introducing distributed interaction (DI) nodes on the contact faces of finite elements. Numerical examples show the effectiveness of the present method to treat the contact interaction between material particles and finite elements. [ABSTRACT FROM AUTHOR]

This paper deals with the introduction of mesh refinement techniques within the non-intrusive patch process. For this, an ad hoc residual based explicit error estimator is built, which is adapted to a multi-scale solution, associated with those non-intrusive mesh refinement technique. Moreover, to reduce the global cost of the process, one introduces an estimate of the convergence error of the non-intrusive algorithm, which allows to reduce the number of iterations. This method is discussed and illustrated in various numerical examples. [ABSTRACT FROM AUTHOR]

The paper proposes a variational approach to model brittle fracture propagation based on zero-thickness finite elements. Similar to the phase-field model for fracture, the problem of a fractured structure is variationally formulated by considering a minimization problem involving bulk and fracture surface energies. With the help of a damage variable used as an additional degree of freedom, the fracture propagates according to the values of the minimizers of the total potential energy. This damage variable is restricted to dimensionally reduced interface elements inserted between element boundaries. Crack opening is predicted when the elastic energy within the interface surface exceeds the critical energy release rate. The solution of the discretized system of equations is performed in a staggered scheme, solving first for the displacement field and then searching for the solution for the updated nodal damage variables. Selected numerical examples, including re-analyses of laboratory tests characterized by rather complex crack paths, are presented to demonstrate the performance of the proposed variational interface model. [ABSTRACT FROM AUTHOR]

Feature-driven topology optimization has been extensively studied in the past decade, and the majority of the works have treated it as the multi-component/void layout optimization problem. A major limitation of this treatment is that the number of components/voids should be defined in advance. To overcome this limitation, this paper presents a novel void feature control method, through which the specified void feature (in any geometric form) could be well-contained by each interior void in the finalized topology design, regardless of the quantity. Numerical stability of this method is discussed and an auxiliary algorithm has been developed to enhance it. Other than that, the proposed method also serves the purpose of void length scale control, which is also a hot issue in the topology optimization field. To be specific, by tailoring the specified void feature size, the void length scale is guaranteed to be larger than that. To the authors’ knowledge, the void length scale control was rarely studied under the level set framework. Through a few numerical case studies, it is proven that the void feature control method is effective while only limited compromise of the structural performance has been observed. [ABSTRACT FROM AUTHOR]

With the availability of multi-material additive manufacturing, topology optimization of a multi-material structure and the selection of candidate materials become increasingly important. This paper aims to develop a new multi-material topology optimization algorithm and propose the guidelines for the selection of candidate materials from the database. The multi-material design variables and their inequality relationship are built on volume fractions of multiple materials within each element. The multi-material design variables are relaxed and multiple floating projection constraints simulate their discrete constraints by pushing design variables towards 0 or 1. Meanwhile, their inequality relationship is enforced by their variation limits. The proposed multi-material topology optimization algorithm can be applied to the compliance minimization problem constrained by a single mass or multiple volumes, as demonstrated in numerical examples. This paper mainly focuses on a single mass constraint, and 2D and 3D numerical examples systematically demonstrate that an optimized design under a mass constraint achieves a lower compliance when more materials appear in the final design simultaneously. Furthermore, we establish an approach to predict the inclusion or exclusion of a material from the final design, and propose the conditions for the co-existence of candidate materials, which guide users to select candidate materials from the database. • A new multi-material topology optimization algorithm is proposed. • The algorithm can be directly applied to a single mass constraint or multiple volume constraints. • The results systematically demonstrate the advantage of the multi-material design. • The approach to predict the occurrence of candidate materials in the final design is developed. • The approach can guide the users to select materials for multi-material additive manufacturing. [ABSTRACT FROM AUTHOR]

To model fiber failures in random fiber networks, we have developed an elastoplastic Timoshenko beam finite element with embedded discontinuities. The method is based on the theory of strong discontinuities where the generalized displacement field is enhanced by a jump. The continuum mechanics formulation accounts for a fracture process zone and a bulk material while retaining traction continuity across the discontinuity. The additional degrees of freedom that are associated with the discontinuity are represented by a midpoint node, which is statically condensed to enable the implementation in commercial software through the user element interface. We propose a quasi-brittle fracture model, where the failure-related deformation is uncoupled from the plastic deformation in the bulk material. To retain the positive definite finite element stiffness matrix of the bulk material, we neglect the fracture-related softening of the discontinuity and employ a modified Newton iteration in the strain softening domain. Our implementation facilitates the integration into commercial finite element software and examples illustrate the robustness of the method. The FORTRAN source code is freely available to benchmark our model. We show that fiber failures contribute to the nonlinear stress–strain response of paper. Together with fiber–fiber bond failures, they can potentially explain the nonlinear stress–strain response of paper and nanopaper. [ABSTRACT FROM AUTHOR]

Bayesian system identification has attracted substantial interest in recent years for inferring structural models based on measured dynamic response from a structural dynamical system. The focus in this paper is Bayesian system identification based on noisy incomplete modal data where we can impose spatially-sparse stiffness changes when updating a structural model. To this end, based on a similar hierarchical sparse Bayesian learning model from our previous work, we propose two Gibbs sampling algorithms. The algorithms differ in their strategies to deal with the posterior uncertainty of the equation-error precision parameter, but both sample from the conditional posterior probability density functions (PDFs) for the structural stiffness parameters and system modal parameters. The effective dimension for the Gibbs sampling is low because iterative sampling is done from only three conditional posterior PDFs that correspond to three parameter groups, along with sampling of the equation-error precision parameter from another conditional posterior PDF in one of the algorithms where it is not integrated out as a “nuisance” parameter. A nice feature from a computational perspective is that it is not necessary to solve a nonlinear eigenvalue problem of a structural model. The effectiveness and robustness of the proposed algorithms are illustrated by applying them to the IASE-ASCE Phase II simulated and experimental benchmark studies. The goal is to use incomplete modal data identified before and after possible damage to detect and assess spatially-sparse stiffness reductions induced by any damage. Our past and current focus on meeting challenges arising from Bayesian inference of structural stiffness serves to strengthen the capability of vibration-based structural system identification but our methods also have much broader applicability for inverse problems in science and technology where system matrices are to be inferred from noisy partial information about their eigenquantities. [ABSTRACT FROM AUTHOR]

This paper introduces an efficient sparse recovery approach for Polynomial Chaos (PC) expansions, which promotes the sparsity by breaking the dimensionality of the problem. The proposed algorithm incrementally explores sub-dimensional expansions for a sparser recovery, and shows success when removal of uninfluential parameters that results in a lower coherence for measurement matrix, allows for a higher order and/or sparser expansion to be recovered. The incremental algorithm effectively searches for the sparsest PC approximation, and not only can it decrease the prediction error, but it can also reduce the dimensionality of PCE model. Four numerical examples are provided to demonstrate the validity of the proposed approach. The results from these examples show that the incremental algorithm substantially outperforms conventional compressive sampling approaches for PCE, in terms of both solution sparsity and prediction error. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Abstract: This paper is the second in this series to develop a numerical homogenization method for heterogeneous media and integrate it with goal-oriented finite element mesh adaptivity. The first paper developed the mathematical formulation of local homogenization. This was done using an appropriate averaging of the Moore–Penrose pseudoinverse of the element stiffness matrix. We also presented numerical verification of this approach and other numerical results. In the current paper, we present four algorithms to compute the Moore–Penrose pseudoinverse that also exploit the sparsity. These are based on QR factorization, a priori knowledge of the null-space, a regularization based characterization, and lastly an iterative algorithm based on proper splittings of matrices. We analyze some of these algorithms in detail for homogenizing hexahedral elements in the molecular base model of densification in Step and Flash Imprint Lithography. [Copyright &y& Elsevier]

Abstract: This paper is the first in this series to develop a numerical homogenization method for heterogeneous media and integrate it with goal-oriented finite element mesh adaptivity. We describe the physical application, Step and Flash Imprint Lithography, in brief and present the mathematical ideas and numerical verification. The method requires the Moore–Penrose pseudoinverse of element stiffness matrices. Algorithms for efficiently computing the pseudoinverse of sparse matrices will be presented in the second paper. The purpose of numerical homogenization is to reduce the number of degrees of freedom, find locally optimal effective material properties, and perform goal-oriented mesh refinement. Traditionally, a finite element mesh is designed after obtaining material properties in different regions. The mesh has to resolve material discontinuities and rapid variations in the solution. In our approach, however, we generate a sequence of coarse meshes (possibly 1-irregular), and homogenize material properties on each coarse mesh element using a locally posed constrained convex quadratic optimization problem. This upscaling is done using the Moore–Penrose pseudoinverse of the linearized fine-scale element stiffness matrices, and a material-independent interpolation operator. Numerical verification is done using a two dimensional conductivity problem with known analytical limit. Finally, we present results for two and three dimensional geometries. The results show that this method uses orders of magnitude fewer degrees of freedom to give fast and approximate solutions of the original fine-scale problem. [Copyright &y& Elsevier]

Abstract: In principle, two approaches are possible for resolving strong material micro-heterogeneity: one approach is to adopt homogenization with the underlying assumption of scale separation, whereas the other approach is to completely resolve the fine scale(s) in a single-scale computation. The point of departure for this paper is a recently proposed algorithm for scale-transition such that the two extreme approaches are bridged in a “seamless” fashion. Numerical homogenization is carried out locally, where needed, based on the relation of the macro-scale mesh diameter to the typical length scale of the subscale structure. Moreover, the macroscale mesh adaptivity is driven by an estimation of discretization errors. In the present paper, we generalize this procedure by introducing two-scale adaptivity, whereby subscale discretization errors are viewed as model errors from the macroscale perspective. Numerical examples, adopting elastic–plastic subscale material properties, illustrate the principle and the effectiveness of the adaptive procedure. [Copyright &y& Elsevier]

Abstract: This paper deals with the use of reduced models for solving some optimal control problems. More precisely, the reduced model is obtained through the modal identification method. The test case which the algorithms is tested on is based on the flow over a backward-facing step. Though the reduction for the velocity fields for different Reynolds numbers is treated elsewhere , only the convection–diffusion equation for the energy problem is treated here. The model reduction is obtained through the solution of a gradient-type optimization problem where the objective function gradient is computed through the adjoint-state method. The obtained reduced models are validated before being coupled to optimal control algorithms. In this paper the feedback optimal control problem is considered. A Riccati equation is solved along with the Kalman gain equation. Additionally, a Kalman filter is performed to reconstruct the reduced state through previous and actual measurements. The numerical test case shows the ability of the proposed approach to control systems through the use of reduced models obtained by the modal identification method. [Copyright &y& Elsevier]

Abstract: In this paper a Geometric T–Ω formulation to solve eddy-current problems on a tetrahedral mesh is presented. When non-simply-connected conducting regions are considered, the formulation requires the so-called thick cuts, while, in the literature, more attention is usually given to the so-called thin cuts. While the automatic construction of thin cuts has been theoretically solved many years ago, no implementation of an algorithm to compute the thick cuts which can be used in practice exists so far. In this paper, we propose how to fill this gap by introducing an algorithm to automatically compute the thick cuts on real-sized meshes, based on a belted tree and a tree–cotree decomposition. The belted tree is constructed by means of a homology computation by exploiting efficient reduction methods. A number of benchmarks are presented to demonstrate the generality and the robustness of the algorithm. A rigorous definition of thick cuts, which necessarily has to rely on cohomology, is presented in addition. [Copyright &y& Elsevier]

Abstract: The so-called node-to-segment (NTS) algorithm is probably the most widely used discretization technique for large deformation contact between surfaces with non-matching meshes. Despite the classical NTS contact element has been available in the literature for a long time, its detailed description appears to be currently disseminated in several papers. Moreover, the classical NTS formulation presents some limitations which are well known but have been the subject of limited investigations. In particular, the algorithm presents severe problems in dealing with some special cases, in which the identification of the master segment related to a slave-node is either ambiguous or impossible. The objectives of this paper are to provide a careful review of the classical formulation, and to propose new techniques for the treatment of the aforementioned special cases. The paper is composed of three parts. In the first one the classical formulation of the NTS algorithm in two dimensions and in the frictionless case is reviewed in detail. The main geometric relationships are presented, and the contact contributions to the stiffness matrix and to the residual vector are derived. The second part is devoted to the treatment of the special cases. Some algorithms specially developed to deal with such cases are illustrated. Finally, the third part presents some numerical examples which show the effectiveness of the proposed algorithms. [Copyright &y& Elsevier]

Abstract: In this paper, we discuss the multiscale computation of the eigenvalue problems for the elastic equations in composite materials. The multiscale finite element method for solving these problems is presented. Numerous numerical results validate the proposed method of this paper. [Copyright &y& Elsevier]

Abstract: This paper presents algorithms for solving structural topology optimization problems with uncertainty in the magnitude and location of the applied loads and with small uncertainty in the location of the structural nodes. The second type of uncertainty would typically arise from fabrication errors where the tolerances for the node locations are small in relation to the length scale of the structural elements. We first review the discrete form of the uncertain loads problem, which has been previously solved using a weighted average of multiple load patterns. With minor modifications, we extend this solution to include loads described by continuous joint probability density functions. We then proceed to the main contribution of this paper: structural optimization under uncertainty in the nodal locations. This optimization problem is computationally difficult because it involves variations of the inverse of the structural stiffness matrix. It is shown, however, that for small uncertainties the problem can be recast into a simpler but equivalent structural optimization problem with equivalent uncertain loads. By expressing these equivalent loads in terms of continuous random variables, we are able to make use of the extended form of the uncertain loads problem presented in the first part of this paper. The optimization algorithms are developed in the context of minimum compliance (maximum stiffness) design. Simple examples are presented. The results demonstrate that load and nodal uncertainties can have dramatic impact on optimal design. For structures containing thin substructures under axial loads, it is shown that these uncertainties (a) are of first-order significance, influencing the linear elastic response quantities, and (b) can affect designs by avoiding unrealistically optimistic and potentially unstable structures. The additional computational cost associated with the uncertainties scales linearly with the number of uncertainties and is insignificant compared to the cost associated with solving the deterministic structural optimization problem. [Copyright &y& Elsevier]

Abstract: In practice, a continuum structure is usually designed to carry the traction applied to the boundary of the structure subject to prescribed displacements imposed on its boundary. The design domains of practical structures are often limited and significantly affect the final optimal design of the structures. Structural boundaries under traction and prescribed displacements should be treated as a zero level set in the level set methods. Firstly, to overcome the limitations of current level set methods and the stopping issue of structural boundary movements, this paper presents a set of new level set based optimization formulae for the optimal design of continuum structures with bounded design domains. A set of new normal velocities required by level set movements is given. A level set based optimization algorithm is developed and implemented with several robust and efficient numerical techniques, which include the level set regularization algorithm, gradient projection algorithm, nonlinear velocity mapping algorithm and return mapping algorithm. Secondly, in order to overcome the difficulty in nucleating holes in the design domain in the level set based optimization method, this paper introduces a new optimization strategy with a small possibility random topology mutations and crossovers. A mixed topology optimization algorithm is implemented and presented for the compliance minimization problems of continuum structures with material volume constraints. The validity and effectiveness of the proposed method are demonstrated with two examples. [Copyright &y& Elsevier]

One-step finite element method (also called inverse approach) is more and more widely used in the automobile industry because of its unique advantages. Initial solution is an essential issue to ensure the success of the non-linear resolution in the implicit static one-step finite element solver. In order to speed up the convergence of the Newton–Raphson iterations, different kinds of initial solution methods are known. These are studied and compared in this paper. Several examples are followed to show their performance and efficiency. However most of these methods are based on geometric considerations like the geometric mapping method or the radial length development method. This kind of geometry based mapping methods could not reflect all aspects of the actual forming process. Therefore, an energy based mapping algorithm was implemented and coupled with the reverse deformation method which is based on the assumption of linear elastic deformation. This novel algorithm is proposed to provide the initial solution of one-step finite element method. For a complicated sheet forming modeling initial solutions obtained by different energy based algorithms coupled with the reverse deformation method are then compared in this paper. The results show that the Desbrun quadratic energy method and the accordant parameterization method combined with the inverse deformation method respectively are universal, efficient and robust initial solution schemes for the one-step finite element method. [Copyright &y& Elsevier]

Abstract: This paper presents solution verification studies applicable to a class of problems involving wave propagation, frictional contact, geometrical complexity, and localized incompressibility. The studies are in support of a validation exercise of a phenomenological screw failure model. The numerical simulations are performed using a fully explicit transient dynamics finite element code, employing both standard four-node tetrahedral and eight-node mean quadrature hexahedral elements. It is demonstrated that verifying the accuracy of the simulation involves not only consideration of the mesh discretization error, but also the effect of the hourglass control and the contact enforcement. In particular, the proper amount of hourglass control and the behavior of the contact search and enforcement algorithms depend greatly on the mesh resolution. We carry out the solution verification exercise using mesh refinement studies and describe our systematic approach to handling the complicating issues. It is shown that hourglassing and contact must both be carefully monitored as the mesh is refined, and it is often necessary to make adjustments to the hourglass and contact user input parameters to accommodate finer meshes. We introduce in this paper the hourglass energy, which is used as an “error indicator” for the hourglass control. If the hourglass energy does not tend to zero with mesh refinement, then an hourglass control parameter is changed and the calculation is repeated. [Copyright &y& Elsevier]

Abstract: During a high velocity impact of a structure on a nearly incompressible fluid, impulse loads with high-pressure peaks occur. This physical phenomenon called ‘slamming’ is a concern in shipbuilding industry because of the possibility of hull damage. Shipbuilding companies have carried out several studies on slamming modeling using FEM software with added mass techniques to represent fluid effects. In the added mass method inertia effects of the fluid are not taken into account and are only valid when the deadrise angle is small. This paper presents the prediction of the local high pressure load on a rigid wedge impacting a free surface, where the fluid is represented by solving Navier–Stokes equations with an Eulerian or ALE formulation. The fluid–structure interaction is simulated using a coupling algorithm; the fluid is treated on a fixed or moving mesh using an ALE formulation and the structure on a deformable mesh using a Lagrangian formulation. A new coupling algorithm is developed in the paper. The coupling algorithm computes the coupling forces at the fluid–structure interface. These forces are added to the fluid and structure nodal forces, where fluid and structure are solved using an explicit finite element formulation. Predicting the local pressure peak on the structure requires an accurate fluid–structure interaction algorithm. The Euler–Lagrange coupling algorithm presented in this paper uses a penalty based formulation similar to penalty contact in Lagrangian analyses. Both penalty coupling and penalty contact can generate high frequency oscillations due to the nearly incompressible nature of the fluid. In this paper, a damping force based on the relative velocity of the fluid and the structure is introduced to smooth out non-physical high frequency oscillations induced by the penalty springs in the coupling algorithm. [Copyright &y& Elsevier]

Abstract: This paper presents a return mapping algorithm for cyclic viscoplastic constitutive models that include material memory effects. The constitutive model is based on multi-component forms of kinematic and isotropic hardening variables in conjunction with von Mises yield criterion. Armstrong–Frederick (A–F) type rules are used to describe the nonlinear evolution of each of the multi-component kinematic hardening variables. A saturation type (exponential) rule is used to describe the nonlinear evolution of each of the isotropic hardening variables. The concept of memory surface is used to describe the strain range dependent material memory effects that are induced by the prior strain histories. In this paper, the above class of cyclic viscoplastic constitutive models is formulated within a consistent thermodynamic framework that encompasses the standard generalized materials framework. Furthermore, a complete algorithmic treatment of the above rate-dependent constitutive model is also presented for any desired stress or strain constrained configuration subspace. A generalized midpoint algorithm is used to integrate the rate constitutive equations. The consistent tangent operator is obtained by linearizing the return mapping algorithm, and is found to be unsymmetric due to the presence of nonlinear evolution rules for the kinematic hardening variables. Several numerical examples representing the cyclic hardening and softening behavior, transient and stabilized hysteresis behavior, and the non-fading memory effects of the material are presented. These examples demonstrate the accuracy and robustness of the present algorithmic framework for modeling the cyclic viscoplastic behavior of the material. [Copyright &y& Elsevier]

Abstract: For a wide class of problems in continuum mechanics like those involving phase transitions or finite elastoplasticity, the governing potentials tend to be not quasiconvex. This leads to the occurrence of microstructures of in principle arbitrarily small scale, which cannot be resolved by standard discretization schemes. Their effective macroscopic properties, however, can efficiently be recovered with relaxation theory. The paper introduces the variational framework necessary for the implementation of relaxation algorithms with emphasis on problems with internal variables in a time-incremental setting. The methods developed are based on numerical approximations to notions of generalized convexification. The focus is on the thorough analysis of numerical algorithms and their efficiency in applications to benchmark problems. An outlook to time-evolution of microstructures within the framework of relaxation theory concludes the paper. [Copyright &y& Elsevier]

This paper presents and evaluates several variationally consistent state determination algorithms for geometric and material nonlinear beam finite elements based on the Hellinger–Reissner (HR) assumed stress variational principle. The specifically targeted element type is derived from the two-field (displacement and generalized stress) form of this principle. Momentum balance and strain compatibility are expressed in a weak-form, and the constitutive equations are satisfied directly at each integration point (i.e., section) along the element length. A specific formulation of one element of this type is presented in a companion paper. Interelement compatibility is not enforced in the dual (stress resultant) variables for this element, and thus various state determination algorithms can be formulated at the local (i.e., element and section) levels. Four state determination algorithms, referred to as the L–L, L–N, N–L and N–N procedures, are considered. The first symbol in these names indicates the element level and the second denotes the section level. Furthermore, the symbol N indicates that local nonlinear iteration is performed prior to returning to the higher level, whereas the symbol L indicates that only the linearized equations are satisfied at the corresponding local level. The L–L algorithm corresponds to conventional Newton iteration on both the element and section levels, whereas the N–N algorithm entails the iterative satisfaction of the nonlinear element compatibility and section constitutive equations for each global iteration. The performance of these algorithms is evaluated and compared. It is found that the algorithms in which the local nonlinear strain–displacement compatibility conditions are solved iteratively at the element level for every global iteration lead to fewer global iterations at the expense of iterations at the element level. In particular, the N–N algorithm, is overall the most efficient of the four algorithms for the problems tested. Also, the N–N algorithm has substantially smaller element storage requirements. The N–N algorithm is expected to be particularly advantageous for cases in which a large number of elements remain elastic and inelasticity is concentrated in a few elements. [Copyright &y& Elsevier]

The paper presents combination of discrete element method (DEM) and finite element method (FEM) for dynamic analysis of geomechanics problems. Combined models can employ spherical (or cylindrical in 2D) rigid elements and finite elements in the discretization of different parts of the system. The FEM is a suitable tool to model soil/rock materials while the FEM in many cases can be a better choice to model other parts of the system considered. A typical example can be an idealization of rock cutting with a tool discretized with finite elements and rock or soil samples modelled with discrete elements. The FEM presented in the paper allows large elasto-plastic deformations in the solid regions. Such problems require the use of stabilized FEM to deal with the incompressibility constraint in order to eliminate the volumetric locking defect, especially when using triangular and tetrahedra elements with equal order interpolation for the displacement and the pressure variables. In the paper a stabilization based on the finite calculus (FIC) approach is used. Both theoretical algorithms of DEM and stabilized FEM are implemented in an explicit dynamic code. The paper presents some details of both formulations. A combined numerical algorithm is described finally. Selected numerical results illustrate the possibilities and performance of discrete/finite element analysis in geomechanics problems. [Copyright &y& Elsevier]

On the theoretical level, the present paper presents a detailed comparison of recent finite strain models for Armstrong–Frederick kinematic hardening. Thereby two strategies are discussed: (1) “Chaboche-type” concepts, considering the back stress as internal variable, (2) continuum mechanical extensions of the classical rheological model, using only strain-like internal variables. It is shown in the paper that models of the second kind can be recast in the format of anisotropic inelasticity with structure tensors. Second, the work focuses on the algorithmic treatment of the kinematic hardening concepts presented before. This problem has been tackled up to now only in the context of linearized models. In contrast to isotropic finite elastoplasticity, the integration cannot be carried out with respect to principal axes. Therefore, a new integration algorithm is developed which is suitable for the anisotropic case but still retains plastic incompressibility. In the case of small elastic deformation, the algorithm reduces to a system of only one non-linear equation and twelve linear equations. In general, the computational effort of the new scheme does not exceed the one of the backward Euler scheme which has the disadvantage that plastic incompressibility is not fulfilled automatically. Several numerical examples show that the representatives of both approaches, (1) and (2), yield similar results, if physically reasonable material parameters are chosen. [Copyright &y& Elsevier]

In this paper, bi-discontinuous time step integration algorithms for Linear second-order equations are investigated. The second-order equations are manipulated directly. Both the initial and final displacement and velocity are only weakly enforced at the beginning and at the end of the time interval. As a result, there may be discontinuous jumps for both the displacement and velocity at the two ends of the time interval. If the displacement is assumed to be a polynomial of degree n+1 within the time interval, there are n+4 unknowns in the formulation. It is found that the order of accuracy would be at least n+1 for displacement and n for velocity in general. It is shown that the relevant algorithmic parameters for the second-order equations can be related to the algorithmic parameters presented in Part 1 of this paper for the first-order equations. Unconditionally stable higher order accurate time step integration algorithms with controllable numerical dissipation equivalent to the generalized Pade´ approximations with an order of accuracy 2n+3 can then be constructed systematically. The weighting parameters and the corresponding weighting functions are given explicitly. It is also shown that the accuracy of the particular solutions is compatible with the accuracy of the homogeneous solutions if the proposed weighting functions are employed. [Copyright &y& Elsevier]

In the companion paper the neural network training procedures used in the present evaluation were described in detail. Also, in the companion paper, two new training methods that used the derivative information to build global approximations, and a mixed local-global approximation scheme were proposed in addition to the common function-based training method. In this paper, we carry out an exhaustive numerical evaluation of the proposed and existing methods using five test problems with varying complexity. Approximately 1100 test cases were run as part of the numerical evaluations. First, the efficiency of three different optimization algorithms used for training including a sequential quadratic programming algorithm, a Levenberg–Marquardt algorithm and a genetic algorithm is evaluated. The developed training schemes were next evaluated by comparing with the classical structural optimization methods. The two new derivative-based training methods were found to be more accurate but less efficient than function-based training. Further, compared with classical structural optimization by non-linear programming algorithms, including design sensitivity analysis, global approximation and mixed local-global approximation using neural networks were found to be far less efficient. With careful training, the neural networks were found capable of predicting the exact solution, but this ability depended considerably on the complexity of the original problem. In general, the accuracy of the network, which is a function of the training effort, suffered considerably in large problems. The most surprising observation of the present study was that the training process required over 90% of the total solution time, even when considerable effort was exercised at using the most efficient training procedure. As expected, the use of neural network approximations (once training was complete) speeded up the optimal design process significantly. [Copyright &y& Elsevier]

Design optimization using approximations based on feed-forward back-propagation neural network is the topic of much recent research. The neural network schemes that have been proposed in the literature for optimal design of structural systems differ in their architecture and training procedures. Furthermore, their utility vis-a-vis classical optimization techniques is not always clear. A systematic comparison of the efficiency and accuracy of the neural network-based solution schemes to classical structural optimization techniques is the aim of this and the companion paper. In this paper, the neural network training procedures used in the present evaluation are described in detail. When using first-order nonlinear programming algorithms with neural networks, the ability to approximate derivatives is important. Therefore, mainly for completeness of evaluation, two new training methods that use the derivative information are proposed in addition to the now common function-based training method. The first method uses the derivatives to create additional training points in the vicinity of the original points, based on Taylor''s series expansion. The second method attempts to minimize the error in derivatives while imposing the error in output functions as constraint. Expressions for analytical derivatives are derived for both function-based and derivative-based training. Significant savings in computational time are reported when calculating derivatives using built-in analytical derivatives instead of using finite difference derivatives. In the companion paper the proposed methods are applied to solve five optimization problems with varying degree of complexity. Approximately 1100 test cases are executed in the companion paper to compare the accuracy and efficiency of neural network-based optimization with the classical approaches. [Copyright &y& Elsevier]

Topology optimization using the variable substitution among three fields can achieve a design with desired solid and/or void features. This paper proposes a three-field floating projection topology optimization (FPTO) method using the linear material interpolation. The implicit floating projection constraint is used as an engine for generating a 0/1 solution at the design field. The substitution filtering and projection schemes enhance the length scale and solid/void features to accelerate the formation of structural topology in the physical field. Meanwhile, the three-field FPTO method can be extended to robust formulation, which obtains the eroded, intermediate, and dilated designs with the same topology. The most distinct feature of the FPTO method lies in the adoption of the linear material interpolation scheme, which makes many topology optimization problems straightforward. As an example, the proposed three-field FPTO algorithm is further applied to the design of shell-infill structures using the linear multi-material interpolation scheme. The distribution of the shell material is generated through a simple filtering scheme, and the shell thickness is accurately controlled by the filter radius. Numerical examples are presented to demonstrate the effectiveness and advantage of the proposed three-field FPTO method. • A three-field floating projection topology optimization method is proposed. • The algorithm can use the linear material interpolation scheme. • The algorithm is extended straightforwardly for robust formulation. • A simple approach for the topology optimization of shell-infill structures is proposed. [ABSTRACT FROM AUTHOR]

The aim of this paper is to analyze the integration for 3D isogeometric finite element method solvers and its effective scheduling on hierarchical computer architecture. Data necessary for concurrency over elements is independent, so computation on this level is trivially concurrent. However, constructing several layers of concurrency for the integration algorithm is challenging. In this work, we propose a multilevel concurrent integration algorithm associated with scheduling that brings one extra degree of possible speedup. Because of one extra degree of possible speedup, we analyze the concurrent integration inside elements. The scheduling algorithm is intended for strongly related hierarchical architectures of a GPU. Using trace theory and Foata Normal Form, we verify integrity of the proposed solution. Summing up, we propose a general method for analyzing concurrency of the integration algorithm. We instantiate this method on a classical element-based integration algorithm, however, this methodology is possible to apply for other integration algorithms, including sum factorization, fast numerical quadrature, or row-wise integration methods. • Multilayer parallel integrating coefficients of IsoGeometric Analysis IGA equations. • Trace theory model used for the formal verification. • Dikert graph based sub-optimal scheduling on cluster with GPU units. • Efficiency and scalability tests of proposed parallel IGA for L2 projection problem. • Application to sum factorization, fast numerical quadrature, row-wise integration. [ABSTRACT FROM AUTHOR]

In numerical analysis the failure of engineering materials is controlled through specifying yield envelopes (or surfaces) that bound the allowable stress in the material. However, each surface is distinct and requires a specific equation describing the shape of the surface to be formulated in each case. These equations impact on the numerical implementation, specifically relating to stress integration, of the models and therefore a separate algorithm must be constructed for each model. This paper presents, for the first time, a way to construct yield surfaces using techniques from non-uniform rational basis spline (NURBS) surfaces, such that any isotropic convex yield envelope can be represented within the same framework. These surfaces are combined with an implicit backward-Euler-type stress integration algorithm to provide a flexible numerical framework for computational plasticity. The algorithm is inherently stable as the iterative process starts and remains on the yield surface throughout the stress integration. The performance of the algorithm is explored using both material point investigations and boundary value analyses demonstrating that the framework can be applied to a variety of plasticity models. [ABSTRACT FROM AUTHOR]