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For linear elastic functionally graded materials (FGMs), the fracture parameters describing the crack tip fields include not only stress intensity factors (SIFs) but also T-stress (nonsingular stress). These two fracture parameters are important for determining the crack initiation angle under mixed-mode loading conditions in brittle FGMs (e.g. ceramic/ceramic such as TiC/SiC). In this paper, the mixed-mode SIFs and T-stress are evaluated by means of the interaction integral, in the form of an equivalent domain integral, in combination with the finite element method. In order to predict the crack initiation angle in brittle FGMs, this paper makes use of a fracture criterion which incorporates the T-stress effect. This type of criterion involves the mixed-mode SIFs, the T-stress, and a physical length scale rc (representative of the fracture process zone size). Various types of material gradations are considered such as continuum models (e.g. exponentially graded material) and micromechanics models (e.g. self-consistent model). Several examples are given to show the accuracy and efficiency of the interaction integral scheme for evaluating mixed-mode SIFs, T-stress, and crack initiation angle. The techniques developed provide a basic framework for quasi-static crack propagation in FGMs. [Copyright &y& Elsevier]

Averaging techniques are popular tools in adaptive finite element methods for numerical simulation in continuum mechanics since they provide efficient a posteriori error control. In this paper, the reliability of any averaging estimator is shown for low order finite element methods in one time-step of elastoplasticity with hardening. The constants and higher-order terms are effected by the hardening and the smoothness of given right-hand sides, but are independent of the structure of a shape-regular mesh. Since it involves a different functional analytical framework, the case of perfect plasticity is excluded from this paper. [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 this paper, we present a boundary-volume based Lippmann Schwinger integral equation for numerical homogenization problems of heterogeneous conductive materials with arbitrary contrast ratio and imperfect interface behavior. It is shown that the interior temperature gradient within each homogeneous phase is connected to the material dissimilarity quantities at all boundaries between different phases. The Kapitza interface model can also be included in the formulation. The basic FFT iteration schemes converge fast even for infinite contrast in the case of usual interface conditions. In the case of Kapitza's interface, Conjugate Gradient schemes based on the new formulation converge fast and yield accurate results when compared with standard Finite Element Method. • Boundary-volume formulation for conductive composites at arbitrary contrast. • Fourier series method accounting for interface discontinuities and singularities. • Fast iterative scheme for the infinite contrast and imperfect interface problems. • Analytical form factors which can be applied to phases of any shapes. [ABSTRACT FROM AUTHOR]

In the present work, enhanced one-dimensional finite elements with node-dependent kinematics are proposed for the static analysis. Finite element governing equations are derived by applying the Carrera Unified Formulation. This framework subdivides the three-dimensional displacement field into a cross-section domain and an axis domain. The dimension along the beam is discretized by using Lagrange-based shape functions. At each node of the element, an independent structural theory can be imposed, thus obtaining a node-dependent kinematic model. This method permits to focus on the finite element node. In this way, several combinations of kinematics can be used together. In particular, Taylor-based and Legendre-based expansions have been adopted in this paper to create global–local models without using special coupling methods. The results have been compared with well-established benchmarks from the literature. Compact section and thin-walled beams have been taken into account. Results have been given in terms of displacements and stresses. It is shown that the present model provides high accuracy with a reduced number of degrees of freedom. [ABSTRACT FROM AUTHOR]

The derivation and implementation of an asynchronous direct time integration scheme for domain-decomposed finite element models is presented. To maximize clarity in the description of the proposed asynchronous integration, the scheme is restricted to the linear-elastic stress wave propagation case. The proposed method allows the integration of individual subdomains with independent time steps. There is no requirement for an integer time steps ratio of the interacting domains while maintaining zero interface energy. The subdomains are connected by the condition of the continuity of the acceleration field at the interface. In addition, the a posteriori technique is applied to satisfy the continuity of the displacement and velocity fields. Another important contribution of this paper lies in the description of the implementation — we offer the reader a general description of the implementation of the case of any number of subdomains with any number of constraints between them, while the basics of the algorithm are explained on a single domain pair. The functionality of the asynchronous integrator is verified by solving selected problems and comparing with analytical solutions and experimental measurements obtained using a Split Hopkinson pressure bar setup. • FEM model is decomposed to subdomains which are individually integrated in time. • Each domain is integrated with individually set time step. • No restriction for ratio of time steps of connected domain pair are imposed. • Continuity of acceleration, velocity and displacement at the interface is satisfied. • Individual domains can be time integrated by different schemes. [ABSTRACT FROM AUTHOR]

In this paper, a number of frequency-domain dynamic analysis procedures of randomly disordered structural systems in the medium frequency range are integrated into the stochastic finite element method. In all cases, frequency-domain model reduction strategies are used to minimize the computational effort in the mid-frequency range. Firstly, an energy operator approach (EOA) is investigated. In this procedure, an energy operator adapted to a fixed medium frequency band is defined whose dominant eigensubspace is used to construct a reduced model using a Ritz–Galerkin method. Secondly, the proper orthogonal decomposition method is used to extract the spatial dominant coherent structures in the vibration wave field in the mid-frequency band. The coherent structures are the eigenvectors corresponding to the dominant eigenvalues of the spatial autocorrelation function of the system response. Consequently, a close relationship between the energy operator approach and the proper orthogonal decomposition method is identified, although these two approaches are not mathematically identical. The proper orthogonal decomposition method appears to be more straightforward compared to the energy operator method from the viewpoint of numerical implementation. Thirdly, another approach, namely the dynamic element method based on frequency-dependent finite element shape functions, is considered. A stochastic reduction method is then utilized to represent the uncertain parameters, modelled as stochastic processes, in terms of their dominant scales of fluctuation. The Karhunen–Loeve and the polynomial chaos decompositions are used to that effect, in the context of a stochastic finite element formalism. The methodology adopted in the paper thus integrates efficient dynamical reduction techniques with a reduction scheme of stochastic processes for the mid-frequency vibration of linear random systems. The formulation is exemplified by its application to the analysis of the dynamics of a coupled uncertain rod assembly subjected to an external excitation. The example is also used to highlight some of the relative features of the three dynamic reduction strategies. [Copyright &y& Elsevier]

Analysis of wave propagation in solid materials under moving loads is a topic of great interest in railway engineering. The objective of the present paper is three-dimensional modelling of high-speed train related ground vibrations; in particular the question of how to account for the unbounded media is addressed.For efficient and accurate modelling of railway structural components taking the unbounded media into account, a hybrid method based on the combination of conventional finite element method and scaled boundary finite element method is established. In the paper, element matrices and solution procedures for the Scaled Boundary Finite Element Method (SBFEM) are derived. A non-linear finite element iteration scheme using Lagrange multipliers and coupling between the unbounded domain and the finite element domain are also discussed. Two numerical examples including one example demonstrating the dynamical response of a railroad section are presented to demonstrate the performance of the proposed method. [Copyright &y& Elsevier]

The finite element method is used to simulate coupled problems, which describe the related physical and chemical processes of ore body formation and mineralization, in geological and geochemical systems. The main purpose of this paper is to illustrate some simulation results for different types of modelling problems in pore-fluid saturated rock masses. The aims of the simulation results presented in this paper are: (1) getting a better understanding of the processes and mechanisms of ore body formation and mineralization in the upper crust of the Earth; (2) demonstrating the usefulness and applicability of the finite element method in dealing with a wide range of coupled problems in geological and geochemical systems; (3) qualitatively establishing a set of showcase problems, against which any numerical method and computer package can be reasonably validated. [Copyright &y& Elsevier]

This article considers a generalization of the classical structural flexibility matrix. It expands on previous papers by taking a deeper look at computational considerations at the substructure level. Direct or indirect computation of flexibilities as “influence coefficients” has traditionally required pre-removal of rigid body modes by imposing appropriate support conditions, mimicking experimental arrangements. With the method presented here the flexibility of an individual element or substructure is directly obtained as a particular generalized inverse of the free–free stiffness matrix. This generalized inverse preserves the stiffness spectrum. The definition is element independent and only involves access to the stiffness generated by a standard finite element program and the separate construction of an orthonormal rigid-body mode basis. The free–free flexibility has proven useful in special application areas of finite element structural analysis, notably massively parallel processing, model reduction and damage localization. It can be computed by solving sets of linear equations and does not require processing an eigenproblem or performing a singular value decomposition. If substructures contain thousands of d.o.f., exploitation of the stiffness sparseness is important. For that case this paper presents a computation procedure based on an exact penalty method, and a projected rank-regularized inverse stiffness with diagonal entries inserted by the sparse factorization process. These entries can be physically interpreted as penalty springs. This procedure takes advantage of the stiffness sparseness while forming the full free–free flexibility, or a boundary subset, and is backed by an in-depth null space analysis for robustness. [Copyright &y& Elsevier]

The present paper investigates the formulation of 3D co-rotational beam elements for the buckling and post-buckling analysis of frame structures. Following Pacoste and Eriksson [Comput. Methods Appl. Mech. Engrg. 144 (1997) 163], the term co-rotational relates here to the provision of a local reference frame that continuously rotates and translates with the element. Within this context, several issues are emphasised. The first one refers to the parameterisation of finite 3D rotations. The alternative put forth in the paper is based on the spatial form of the incremental rotational vector. The second issue concerns warping effects which are introduced by adding a seventh degree of freedom at each node. Different types of local formulations are considered and it is shown that at least some degree of non-linearity must be introduced in the local strain definition in order to obtain correct results for certain classes of problems. Within the present approach the centroid and shear center of the cross-section are not necessarily coincident. Finally, in the context of instability problems, a method for the direct computation of critical points is also briefly discussed. This is based on a minimal augmentation procedure as developed by Eriksson [Comput. Methods Appl. Mech. Engrg. 156 (1998) 45; Comput. Methods Appl. Mech. Engrg. 179 (1999) 265; Int. J. Struct. Stability Dynamic 1 (1) (2001)]. Ten examples, including large displacement and stability problems, are used in order to assess the performances of the elements. [Copyright &y& Elsevier]

This paper presents a new stochastic finite element method for computing structural stochastic responses of linear problems. The method provides a new expansion of stochastic response and decouples the stochastic response into a combination of a series of deterministic responses with random variable coefficients. A dedicated iterative algorithm is proposed to compute the deterministic responses and corresponding random variable coefficients one by one. The algorithm computes the deterministic responses and corresponding random variable coefficients in their individual spaces and is insensitive to stochastic dimensions, thus it can be applied to high-dimensional stochastic problems without extra difficulties. More importantly, the deterministic responses can be computed efficiently by use of existing finite element method (FEM) solvers, thus the proposed method can be easy to embed into existing FEM structural analysis software. Three practical examples, including low-dimensional and high-dimensional stochastic problems, are given to demonstrate the accuracy and effectiveness of the proposed method. • This paper develops a new method for solving stochastic finite element equations. • Computational costs are low for large-scale and high-dimensional stochastic problems. • The Curse of Dimensionality of high dimensions can be avoided with great success. [ABSTRACT FROM AUTHOR]

In this paper a Powell–Sabin finite-element (PS-FEM) scheme is presented for the solution of the 2D Euler equations in supersonic regime. The spatial discretization is based on PS splines, that are piecewise quadratic polynomials with a global C 1 continuity, defined on conforming triangulations. Some geometrical issues related to the practical construction of the PS elements are discussed, in particular, the generation of the control triangles and the imposition of the boundary conditions. A stabilized formulation is considered, and a novel shock-capturing technique in the context of continuous finite-elements is proposed to reduce oscillations around the discontinuity, and compared with the classical technique proposed by Tezduyar and Senga (2006). The code is verified using manufactured solutions and validated using two challenging numerical examples, which allows to evaluate the performance of the PS discretization in capturing the shocks. [ABSTRACT FROM AUTHOR]

This paper presents the Virtual Element Method (VEM) for the modeling of crack propagation in 2D within the context of linear elastic fracture mechanics (LEFM). By exploiting the advantage of mesh flexibility in the VEM, we establish an adaptive mesh refinement strategy based on the superconvergent patch recovery for triangular, quadrilateral as well as for arbitrary polygonal meshes. For the local stiffness matrix in VEM, we adopt a stabilization term which is stable for both isotropic scaling and ratio. Stress intensity factors (SIFs) of a polygonal mesh are discussed and solved by using the interaction domain integral. The present VEM formulations are finally tested and validated by studying its convergence rate for both continuous and discontinuous problems, and are compared with the optimal convergence rate in the conventional Finite Element Method (FEM). Furthermore, the adaptive mesh refinement strategies used to effectively predict the crack growth with the existence of hanging nodes in nonconforming elements are examined. [ABSTRACT FROM AUTHOR]

Within the novel approach for structural health monitoring of civil engineering structures based on smart self-sensing structural materials, Carbon NanoTube (CNT)-reinforced cement-based composites, often termed “smart concretes”, have drawn rising interest. These composites exhibit self-sensing capabilities resulting in measurable variations of their electrical properties under applied mechanical deformations. This, along with the similarity between these composites and conventional structural concrete, makes it possible to devise cost-efficient distributed monitoring systems for large-scale Reinforced Concrete (RC) structures. While several papers in the literature have focused on pilot applications of smart concrete strain sensors embedded into large RC components, the response of these sensors is not yet fully understood at the fundamental physical level and, as a consequence, their output is not properly interpreted. In order to shed some light on this issue, previous work by the authors focused on laterally unconstrained uni-axial loading conditions. This work extends the previous studies by presenting a mixed micromechanics and finite element approach for the analysis of CNT-reinforced composites subjected to arbitrary strain states. The two mechanisms that contribute to the electrical conductivity of CNT-reinforced composites, namely electron hopping and conductive networking, are contemplated within a percolation framework in the micromechanics model. On the basis of the micromechanics model, the 3D piezoresistivity matrix is determined, for the first time in the literature, by means of virtual dilation and distortion tests. Afterwards, the electro-mechanical modeling of three-dimensional composite elements is conducted by a multi-physics finite element code. The potential of the presented approach is illustrated by extensive parametric analyses, as well as a comparison against experimental data, including application of the mixed micromechanics-finite-element multi-physics formulation for electro-mechanical modeling of three-dimensional elements. [ABSTRACT FROM AUTHOR]

Symmetry and repetitive patterns are very common in both natural and artificial structures. Design with symmetry and pattern repetition (SPR) can be very useful in practical engineering. In this paper, we discuss a new optimal structural design method considering SPR constraints. The basic idea consists in combining a parameterized level-set method together with radial basis function (RBF) interpolation strategy, which allows for decoupling RBF knots from the finite element (FE) analysis mesh. The merits of the decoupled manner are that RBF knots can be mapped independently on regular or irregular design domain, and SPR constraints can be imposed by simply building a bijective relationship between RBF knots regardless of analysis mesh. Furthermore, due to parameterization of the level-set function, there are fewer numerical manipulations that need to be implemented than the conventional level-set method during the optimization process. Several numerical examples are presented to show feasibility and effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

This paper proposes a numerical method for determining the evolution of lubricant film thickness and pressure in partially and fully flooded regions of a hydrodynamic contact between two non-conformal rigid surfaces. The proposed method accounts for the classical Reynolds equation within the fully flooded region and for film separation with surface tension driven flow in the partially flooded region, while at the same time it resolves the a priori unknown boundary between the two regions. Additionally it deals with transitions between wetted, partially flooded regions to dry regions, where the film thickness is zero. Both pressure and film thickness fields are considered as unknowns to solve for in each time step and they are approximated through quadratic B-spline finite elements. The geometry of the gap between the rigid surfaces delimiting the lubricant is accounted for in the form of a unilateral contact condition. Appropriate complementarity conditions with respect to separation or no penetration and no slip between the lubricant and the rigid surfaces are enforced by means of a weighted residual formulation. [ABSTRACT FROM AUTHOR]

This paper introduces a hybrid a posteriori error estimator for the conforming finite element method, which may be regarded as a combination of the explicit residual and the improved ZZ error estimators. With comparable cost, the hybrid estimator is more accurate than the residual estimator. It is shown that the hybrid estimator is reliable on all meshes, unlike estimators of the ZZ type. Moreover, the reliability constant is independent of the jump of the diffusion coefficients for elliptic interface problems under the monotonicity assumption of the coefficients. Finally, numerical examples confirm the robustness of the estimator with respect to coefficient jumps and also better effectivity index compared to the residual estimator. [ABSTRACT FROM AUTHOR]

The Local Average Contact (LAC) method allows the handling of nonmatching meshes in an easy way by averaging locally the interpenetration between the contacting bodies. In this paper we consider the frictionless unilateral contact problem and several numerical experiments involving two and three-dimensional bodies discretized with various linear and quadratic finite elements. We also present a convergence analysis of the method in the geometrical nonconforming case in which the boundary points of the candidate contact areas do not coincide. [ABSTRACT FROM AUTHOR]

Evidence theory is widely regarded as a promising mathematical tool for epistemic uncertainty analysis. However, the heavy computational burden has severely hindered its application in practical engineering problems, which is essentially caused by the discrete uncertainty quantification mechanism of evidence variables. In this paper, an efficient epistemic uncertainty analysis method using evidence theory is proposed, based on a probabilistic and continuous representation of the epistemic uncertainty presented in evidence variables. Firstly, each evidence variable is equivalently transformed to a Johnson p-box which is a family of Johnson distributions enveloped by the CDF bounds. Subsequently, the probability bound analysis is conducted for the input Johnson p-box and the response CDF based on monotonicity analysis. Finally, the CDF bounds of the response are directly calculated using the CDF bounds of the input Johnson p-boxes, by which a high computational efficiency is achieved for the proposed method. Two mathematical problems and two engineering applications are presented to demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

An adaptive multiscale approach for identifying multiple flaws in structures is proposed in this paper. The approach includes a two-step process in which a coarse-scale search initially identifies the candidate subdomains and then a fine-scale search of the narrow subdomains captures the true flaws. The extended finite element method (XFEM) is employed to solve the forward problem because it does not require re-meshing in each iteration. A discrete artificial fish swarm (DAFS) algorithm with adaptive vision is adopted to accelerate the convergence of the objective function. The probable subdomains are output via K-means clustering with an initial estimated number of flaws Then, the artificial bee colony (ABC) algorithm is employed to capture the true flaws and remove the false flaws. The DAFS algorithm is a single-population algorithm, while the ABC algorithm is a multi-population algorithm. Three numerical examples are studied to evaluate the accuracy and performance of the proposed approach. Circular and elliptical flaws and the effects of measurement noise are considered to test the robustness. The results show that this approach can effectively quantify and identify multiple internal flaws without any prior knowledge of their quantity. Furthermore, the strategy is more efficient than previously proposed approaches. [ABSTRACT FROM AUTHOR]

This paper presents a new isogeometric topology optimization (TO) method based on moving morphable components (MMC), where the R-functions are used to represent the topology description functions (TDF) to overcome the C 1 discontinuity problem of the overlapping regions of components. Three new ersatz material models based on uniform, Gauss and Greville abscissae collocation schemes are used to represent both the Young’s modulus of material and the density field based on the Heaviside values of collocation points. Three benchmark examples are tested to evaluate the proposed method, where the collocation schemes are compared as well as the difference between isogeometric analysis (IGA) and finite element method (FEM). The results show that the convergence rate using R-functions has been improved in a range of 17%–60% for different cases in both FEM and IGA frameworks, and the Greville collocation scheme outperforms the other two schemes in the MMC-based TO. [ABSTRACT FROM AUTHOR]

This paper focuses on the problem of combined state/parameter estimation in dynamical systems with many degrees of freedom, for instance finite element models, using measured data from the system and nonlinear Bayesian filtering techniques for the estimation. A highly efficient nonlinear Kalman filtering technique is developed, based on a combination of linear or extended Kalman filtering for state estimation and unscented filtering for parameter learning. Such a combination is implemented by applying the principle of marginalization to the unscented transform. This method leads to a very efficient state/parameter filtering algorithm by taking advantage of the fact that Jacobians of the system equations with respect to the dynamic states, required in extended Kalman filtering, can be easily related to outputs of the finite element analysis required for forward propagation. For linear dynamical systems this approach yields an algorithm with identical accuracy as the UKF but reduced computational time, as demonstrated on several medium-size examples. For nonlinear systems, the resulting algorithm is also superior to the UKF in terms of computational time, due to the fact that Jacobians are functions of the tangent stiffness matrix of the system, whose computation is required for propagation in finite element analysis. It is also shown that, even though this algorithm relies on linearization for propagation of moments of the dynamic states, this reduction in accuracy compared to a generic UKF does not affect learning of the parameters for the systems considered herein. [ABSTRACT FROM AUTHOR]

This paper is devoted to the approximation of the convection–diffusion–reaction equation using a mixed, first-order, formulation. We propose, and analyse, a stabilised finite element method that allows equal order interpolations for the primal and dual variables. This formulation, reminiscent of the Galerkin least-squares method, is proven stable and convergent. In addition, a numerical assessment of the numerical performance of different stabilised finite element methods for the mixed formulation is carried out, and the different methods are compared in terms of accuracy, stability, and sharpness of the layers for two different classical test problems. [ABSTRACT FROM AUTHOR]

In this paper, a local–global model reduction method is presented to solve stochastic optimal control problems constrained by stochastic partial differential equations (stochastic PDEs). If the optimal control problems involve uncertainty, we need to use a few random variables to parameterize the uncertainty. The stochastic optimal control problems require solving coupled optimality system for a large number of samples in the stochastic space to quantify the statistics of the system response and explore the uncertainty quantification. Thus the computation is prohibitively expensive. To overcome the difficulty, model reduction is necessary to significantly reduce the computation complexity. We exploit the advantages from both reduced basis method and Generalized Multiscale Finite Element Method (GMsFEM) and develop the local–global model reduction method for stochastic optimal control problems with stochastic PDE constraints. This local–global model reduction can achieve much more computation efficiency than using only local model reduction approach and only global model reduction approach. We recast the stochastic optimal problems in the framework of saddle-point problems and analyze the existence and uniqueness of the optimal solutions of the reduced model. In the local–global approach, most of computation steps are independent of each other. This is very desirable for scientific computation. Moreover, the online computation for each random sample is very fast via the proposed model reduction method. This allows us to compute the optimality system for a large number of samples. To demonstrate the performance of the local–global model reduction method, a few numerical examples are provided for different stochastic optimal control problems. [ABSTRACT FROM AUTHOR]

This paper aims at designing an observer-based feedback law which locally stabilizes the solution to the two dimensional Navier–Stokes equations with mixed boundary conditions. We consider a finite number of controls acting on a portion of the boundary through Robin boundary conditions and construct a linear Luenberger observer based on the point observations of the linearized Navier–Stokes equations. The sensor location for the point observations is determined by the response of feedback functional gains. We prove that the nonlinear system coupled with the observer through the feedback law is locally exponentially stable. Numerical experiments based on a Taylor–Hood finite element method are presented to illustrate the design for different Reynolds numbers. [ABSTRACT FROM AUTHOR]

Industrials have an intensive use of numerical simulations in order to avoid physical testing and to speed up the design stages of their products. The numerical testing is indeed quicker to set-up, less expensive, and supplies a lot of information about the system under study. Moreover, it can be much closer to the physical tests as the computation power increases. Despite the rise of this power, time consuming simulations remain challenging to be used in design process, especially in an optimization study. Crash simulations belong to this category. These rapid dynamic computations are used by RENAULT during the sizing of the vehicle structure in order to ensure that it meets specifications set up to reach safety criteria in case of accidents. They are completed using finite element software such as VPS (Virtual Performance Solver) developed by ESI group that will be used in this study. For car manufacturers, the goal of the optimization study is to minimize the mass of the vehicle (and thus its consumption) by modifying the thicknesses of some parts (from 20 to 100 variables). Industrials such as RENAULT currently perform optimization studies based on numerical design of experiments. The number of computations required by this technique is from 3 to 10 times the number of variables. This is too much in order to be intensively used in a design process. In order to decrease the time-to-market and to explore alternative technical solutions, we explore the potential of using a parametrized reduced order model in the optimization studies. The parametrized reduced order model gives an estimation of the high-fidelity result for a new set of parameters without using the solver, by analysing the existing results of previous computations with various sets of parameters. The developed reduced order model is called ReCUR. It is partly based on a CUR approach embedded in a regression analysis. The regression statistical model uses the data of a few calculations made with the solver. Other tools such as clustering and linear programming are used to get the regression analysis more efficient. It is hoped to drastically reduce the number of required simulations of a standard optimization study. In this paper, the construction of the reduced order model will be presented. Then, the relevancy of using the reduced order model into a design process will be exhibited through the treatment of two industrial test-cases. Some improvements of the method as well as several potential uses will then be outlined. The applications will highlight the promising power of the method to shorten design process using optimization and long-run simulations. [ABSTRACT FROM AUTHOR]

This paper presents an approach for automatically generating high-quality high-order parameterizations for isogeometric analysis on triangulations. A B-spline represented boundary geometry is parameterized into a collection of high-order Bézier triangles or tetrahedra in 2D and 3D spaces, respectively. Triangular Bézier splines are used to represent both the geometry and physical fields over the triangulation. By imposing continuity constraints on the Bézier ordinates of the elements, a set of global C r smooth basis is constructed and used as the basis for analysis. To ensure high quality of the parameterization, both the parametric and physical meshes are optimized to reduce the shape distortion of the high-order elements relative to well-defined reference elements. The shape distortion is defined based on the Jacobian of the triangular Bézier splines, and its sensitivity with respect to the location of control points is derived analytically and evaluated efficiently. Moreover, a sufficient condition is derived to guarantee the generated mesh is free of local self-intersection, thanks to the convex hull property of triangular Bézier splines. By using a Heaviside projection function, the non-negative Jacobian determinant constraints are formulated efficiently as a single optimization constraint. Several 2D and 3D numerical examples are presented to demonstrate that high-quality high-order elements are generated using our approach. [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]

A novel constitutive framework suitable for material interfaces undergoing large deformations in a geometrically exact setting was developed in Ottosen et al. (2016). In contrast to previous works, it permits the description of arbitrary material anisotropies by fulfilling all fundamental balance laws in physics as well as the principle of material objectivity. This paper deals with an efficient finite element implementation of the aforementioned framework in terms of the natural basis vectors. To be more precise, a different, more compact and more direct derivation of this framework is outlined first. It relies on the variational structure of the underlying problem. Subsequently, the aforementioned finite element approximation is elaborated which is finally embedded into a computational homogenization scheme. This scheme allows the analysis of the influence of the novel interface model on the resulting macroscopic (effective) material response. It is shown by numerical examples that the interaction of bulk energies and interface energies leads, in a very natural manner, to a complex size effect. It includes the frequently observed “the smaller the stiffer” relation, but also the less often observed “the smaller the softer” relation. However, since the overall response is usually a superposition of such relations, the effective properties cannot generally be characterized by one of the aforementioned limiting relations. [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]

The main aim of the work presented in this paper was treatment of damage and deformation localisation observed in the finite element method (FEM) analysis of strain softening materials combined with local constitutive models where damage is represented using continuum damage mechanics (CDM). The CDM/FEM approach typically suffers from a number of shortcomings, including mathematical (change of the type of partial differential equations leading to ill-posed boundary value problem), numerical (pronounced mesh dependency) and physical (infinitely small softening zone with the zero dissipated energy). The approach proposed here is still based on the local constitutive model including damage, but introduces an alternative representation of damage effects in the system of linear momentum balance equations. The damage effects are included through equivalent damage force (EDF), which contributes to the right-hand side of the momentum balance equations. The main advantages of this approach are that the problem remains well posed, as the type of partial differential equations remains unchanged when the material enters softening; numerical stability is preserved without a need for regularisation measures; and significantly reduced mesh dependency. In addition, the EDF approach can be used in combination with existing local CDM damage models and does not violate symmetry of the material stiffness tensor. The EDF approach is applicable to modelling of strain softening typically observed in damaged quasi brittle materials such as fibre reinforced composites and concrete. The EDF model was implemented in the in-house developed coupled FEM-SPH code, where an explicit FEM code is coupled with a stable Total-Lagrange form of SPH. Its performance is demonstrated in the analysis of a dynamic one dimensional (1D) stress wave propagation problem, which was analytically solved by Bazant and Belytschko in 1985. For a range of loading rates that correspond to the material softening regime, the numerical results shown nonlocal character with a finite size of the damaged zone, controlled with the damage characteristic length, which can be experimentally determined and is an input parameter independent of the discretisation density. [ABSTRACT FROM AUTHOR]

This paper investigates a new integrated optimization method for laminated composite structures with piezoelectric patch actuators. Three kinds of design variables, i.e. the actuators’ locations, electrical voltages applied on piezoelectric patches and host structural pseudo-densities, are simultaneously optimized to improve the overall deformation precision. To freely place piezoelectric actuator on the host structure surface regardless of coincident finite element in classic lamination theory (CLT), the multi-point constraints (MPC) method is used here to simulate the perfect bonding between actuator layer and host layer, which inherits the advantages of avoidance of remeshing process analytical sensitivity as well as efficient lamination description during the movement of actuators in optimization iterations. For applying to spatially complex shape control, a modified shape error function based on the relative error between computed and desired surfaces is used. The finite circle method (FCM) is implemented to prevent the overlaps among the actuators and those between actuators and boundaries of the global design domain. Finally, numerical examples with plane or curved host structure are tested and discussed to demonstrate the validity and efficiency of this method. [ABSTRACT FROM AUTHOR]

A stable basis for isogeometric analysis of trimmed models is obtained by combining extended B-splines with truncated hierarchical B-splines. While extended B-splines guarantee that the condition number of system matrices is independent of the location of a trimming curve, local refinement is used to improve the robustness of the procedure and the accuracy of the numerical results. The present extended B-spline construction works in the context of Galerkin and collocation methods. The paper focuses on the latter and introduces a new collocation scheme for truncated hierarchical B-splines. A proper transition between refinement levels is assured by a novel balancing algorithm that employs a simple criterion. The enhanced performance of the locally refined stabilization is verified by scalar Laplace and linear elasticity problems analyzed by a collocation based isogeometric boundary element method. The proposed approach yields excellent results and requires few refinement levels to improve the stabilization procedure and accuracy along trimming curves. [ABSTRACT FROM AUTHOR]

INTERNODES is a general method to deal with non-conforming discretizations of second order partial differential equations on regions partitioned into two or several subdomains. It exploits two intergrid interpolation operators, one for transferring the Dirichlet trace across the interface, the others for the Neumann trace. In every subdomain the original problem is discretized by the finite element method, using a priori non-matching grids and piece-wise polynomials of different degree. In this paper we provide several interpretations of the method and we carry out its stability and convergence analysis, showing that INTERNODES exhibits optimal convergence rate with respect to the finite element sizes. Finally we propose an efficient algorithm for the solution of the corresponding algebraic system. [ABSTRACT FROM AUTHOR]

Material point methods suffer from volumetric locking when modelling near incompressible materials due to the combination of a low-order computational mesh and large numbers of material points per element. However, large numbers of material points per element are required to reduce integration errors due to non-optimum placement of integration points. This restricts the ability of current material point methods in modelling realistic material behaviour. This paper presents for the first time a method to overcome finite deformation volumetric locking in standard and generalised interpolation material point methods for near-incompressible non-linear solid mechanics. The method does not place any restriction on the form of constitutive model used and is straightforward to implement into existing implicit material point method codes. The performance of the method is demonstrated on a number of two and three-dimensional examples and its correct implementation confirmed through convergence studies towards analytical solutions and by obtaining the correct order of convergence within the global Newton–Raphson equilibrium iterations. In particular, the proposed formulation has been shown to remove the over-stiff volumetric behaviour of conventional material point methods and reduce stress oscillations. It is straightforward to extend this approach to other material point methods and the presented formulation can be incorporated into all existing material point methods available in the literature. [ABSTRACT FROM AUTHOR]

The paper presents an efficient and accurate algorithm to analyze the fluid/solid and solid/fluid phononic crystals (PCs) with fluid structure interaction (FSI). In the proposed mass-redistributed finite element method (MR-FEM), the dispersion error in the longitudinal wave of fluid domain is minimized by re-distributing the mass elements in the mass matrix to “tune” the balance, while the lumped mass matrix is employed to analyze the elastic wave within the solid. In addition, different Bloch boundary conditions including the square and triangular lattice using MR-FEM are also formulated in this work. The accuracy of the proposed MR-FEM is verified by using the commercial software COMSOL, and the computational efficiency of MR-FEM is significantly greater than the conventional finite element methods (FEM) based on six different numerical examples. The MR-FEM with excellent features has a great potential to extend 3D multi-physics domain of PCs. [ABSTRACT FROM AUTHOR]

This paper proposes a new node-to-node (NTN) scheme for modeling two-dimensional contact problems within a scaled boundary finite element method (SBFEM) framework. The analysis is formulated and solved directly as a mathematical programming problem known as a mixed complementarity problem (MCP). Polygon elements with arbitrary number of edges and nodes are constructed in the SBFEM modeling. Only the edges of these polygon elements are discretized thereby accommodating highly non-uniform discretization. Non-matching meshes are easily converted into matching meshes by appropriate node insertions, thereby allowing the use of the advantageous node-to-node contact scheme. Contact constraints of non-penetration and friction are described directly in a complementarity format that enables, for instance, the satisfaction of non-penetration without having to resort to user-specified control parameters. Numerical examples using 1st and 2nd order scaled boundary elements are presented to highlight the quality of the elements used through satisfaction of various patch tests, and the accuracy and robustness of the overall approach. [ABSTRACT FROM AUTHOR]

Engineering design via CAD software relies on Non-Uniform Rational B-Splines (NURBS) as a means for representing and communicating geometry. Therefore, in general, a NURBS description of a given design can be considered the exact description. The development of isogeometric methods has made the geometry available to analysis methods Hughes et al. (2005). Isogeometric analysis has been particularly successful in structural analysis; one reason being the wide-spread use of two-dimensional finite elements in this field. For fluid dynamics, where three-dimensional analysis is usually indispensable, isogeometric methods are more complicated, yet of course not impossible, to apply in a general fashion. This paper describes a method that enables the solution of fluid–structure-interaction with a matching spline description of the interface. On the structural side, the spline is used in an isogeometric setting. On the fluid side, the same spline is used in the framework of a NURBS-enhanced finite element method (extension of Sevilla et al. (2011)). The coupling of the structural and the fluid solution is greatly facilitated by the common spline interface. The use of the identical spline representation for both sides permits a direct transfer of the necessary quantities, all the while still allowing an adjusted, individual refinement level for both sides. [ABSTRACT FROM AUTHOR]

In many engineering problems, it is customary to use viscoelastic materials in the passive control of structural vibration and noise radiation. The most commonly used numerical tool to obtain detailed information on the performance of complex structures is the finite element (FE) method. However, structures with viscoelastic materials often require very large-scale FE computational models to obtain reliable predictions, which makes original full-order system evaluation intractable due to memory and time limitations. In order to alleviate these problems, model order reduction techniques have become indispensable. Most of them, however, often do not handle complex-valued and especially frequency-dependent system matrices well. Due to the frequency dependency of the material properties, the FE equations of motion for viscoelastic models are not of a standard second-order form as that for regular elastic FE models. In this paper, a new transformation technique based on Taylor’s theorem is introduced to treat the frequency-dependent shear modulus. After transforming the problem into a second-order system equation with remainder term, an adaptive second-order Arnoldi method is applied for the reduction of the computational complexity of structures, incorporating classical free- and constrained-layer damping treatments. In support, a relative error indicator is developed to iteratively enrich the reduced model and determine the position of expansion points used in next step. Three widely used models, the Golla–Hughes–McTavish model, Generalized Maxwell model and Fractional Derivative model for describing the frequency-dependent property of viscoelastic materials are investigated in order to demonstrate the simplicity, versatility and efficiency of the proposed approach. [ABSTRACT FROM AUTHOR]

This paper presents an adaptive refinement algorithm of a B-spline based finite-element approximation of the streamfunction formulation for the large scale wind-driven ocean currents. In particular, we focus on a posteriori error analysis of the simplified linear model of the stationary quasi-geostrophic equations, namely the Stommel–Munk model, which is the fourth-order partial differential equation. The analysis provides a posteriori error estimator for the local refinement of the Nitsche-type finite-element formulation. Numerical experiments with several benchmark examples are performed to test the capability of the posteriori error indicator on rectangular and L -shape geometries. [ABSTRACT FROM AUTHOR]

This paper presents the construction of novel stabilized finite element methods in the convective–diffusive context that exhibit correct-energy behavior. Classical stabilized formulations can create unwanted artificial energy. Our contribution corrects this undesired property by employing the concepts of dynamic as well as orthogonal small-scales within the variational multiscale framework (VMS). The desire for correct energy indicates that the large- and small-scales should be H 0 1 -orthogonal. Using this orthogonality the VMS method can be converted into the streamline-upwind Petrov–Galerkin (SUPG) or the Galerkin/least-squares (GLS) method. Incorporating both large- and small-scales in the energy definition asks for dynamic behavior of the small-scales. Therefore, the large- and small-scales are treated as separate equations. Two consistent variational formulations which depict correct-energy behavior are proposed: (i) the Galerkin/least-squares method with dynamic small-scales (GLSD) and (ii) the dynamic orthogonal formulation (DO). The methods are presented in combination with an energy-decaying generalized- α time-integrator. Numerical verification shows that dissipation due to the small-scales in classical stabilized methods can become negative, on both a local and a global scale. The results show that without loss of accuracy the correct-energy behavior can be recovered by the proposed methods. The computations employ NURBS-based isogeometric analysis for the spatial discretization. [ABSTRACT FROM AUTHOR]

Efficient incompressible flow simulations, using inf–sup stable pairs of finite element spaces, require the application of efficient solvers for the arising linear saddle point problems. This paper presents an assessment of different solvers: the sparse direct solver UMFPACK, the flexible GMRES (FGMRES) method with different coupled multigrid preconditioners, and FGMRES with Least Squares Commutator (LSC) preconditioners. The assessment is performed for steady-state and time-dependent flows around cylinders in 2d and 3d. Several pairs of inf–sup stable finite element spaces with second order velocity and first order pressure are used. It turns out that for the steady-state problems often FGMRES with an appropriate multigrid preconditioner was the most efficient method on finer grids. For the time-dependent problems, FGMRES with LSC preconditioners that use an inexact iterative solution of the velocity subproblem worked best for smaller time steps. [ABSTRACT FROM AUTHOR]

In this paper, an extended finite element method is presented for simulation of interaction between hydraulic fracturing and natural fractures in saturated porous media. The well-known u − p formulation is employed in order to obtain the fully coupled set of governing equations. Natural faults are modeled for both opening and closure modes where the fluid inflow and contact conditions are considered at the interface, respectively. The Darcy law is employed in conjunction with an aperture dependent permeability for the fracture channel to describe the interfacial inflow. The contact constraints of both the solid and fluid phases are imposed using the Penalty method. The Heaviside and modified ramp enrichment functions are introduced to model the required strong and weak discontinuities in the displacement and pressure fields, respectively. Furthermore, new enrichment functions are introduced to model the junction and/or intersection of the discontinuities. The system of nonlinear equations is solved using the Newton–Raphson iterative algorithm with an implicit time stepping scheme. Finally, several numerical examples are chosen to illustrate the robustness of the proposed formulation as well as to study the interaction of hydraulic fracturing with natural faults. [ABSTRACT FROM AUTHOR]

This paper revisits stiffness optimization of non-linear elastic structures. Due to the non-linearity, several possible stiffness measures can be identified and in this work conventional compliance, i.e. secant stiffness designs are compared to tangent stiffness designs. The optimization problem is solved by the method of moving asymptotes and the sensitivities are calculated using the adjoint method. For the tangent cost function it is shown that although the objective involves the third derivative of the strain energy an efficient formulation for calculating the sensitivity can be obtained. Loss of convergence due to large deformations in void regions is addressed by using a fictitious strain energy such that small strain linear elasticity is approached in the void regions. A well posed topology optimization problem is formulated by using restriction which is achieved via a Helmholtz type filter. The numerical examples provided show that for low load levels, the designs obtained from the different stiffness measures coincide whereas for large deformations significant differences are observed. [ABSTRACT FROM AUTHOR]

The paper presents an effective and accurate way to couple FEM meshes to peridynamic grids and uses it to solve 1D, 2D and 3D problems in static and dynamic conditions. Crack propagation with crack branching problems is also solved. An adaptive algorithm is presented which allows to transform FEM nodes into peridynamic nodes. In this way a new computational technique is produced which equips standard FEM with the capability to effectively and efficiently describe 3D problems with crack propagation phenomena. [ABSTRACT FROM AUTHOR]

Simulation of coastal water waves requires a relevant mathematical model and a proper numerical technique which can be used in irregular geometries. In this paper, we study the numerical solution to a class of irrotational Green–Naghdi (GN) equation using the hybridized discontinuous Galerkin method (HDG). We first use Strang splitting to decompose this problem into a hyperbolic and a dispersive sub-problem. For the hyperbolic sub-problem, we use an explicit HDG with Lax–Friedrichs numerical fluxes. We also include different types of boundary conditions in our formulation. Similar to other HDG implementations, our global unknowns are the numerical traces, i.e. the unknowns on the mesh skeleton. This choice can significantly reduce the number of global degrees of freedom. Solving this part of the problem involves Newton iterations at each time step. However, unlike the implicit versions of HDG, the element equations are decoupled and we do not need to solve a large global system of equations. On the other hand, for the dispersive sub-problem we need to solve a global system of equations, implicitly. Since HDG, only involves unknowns on the mesh skeleton, it can be a very efficient alternative for this implicit solve step. By coupling these two solvers, we obtain the solution to the Green–Naghdi equation. Next, we use this technique to solve a set of verification and validation examples. In the first example we show the convergence properties of the numerical method. Next, we compare our results with a set of experimental data for the nonlinear water waves in different situations. Finally, we present a two dimensional simulation result for a solitary wave passing over a cosine bump. We observe close to optimal convergence rates and a good agreement between our numerical results and the experimental data. [ABSTRACT FROM AUTHOR]

Recently, Isogeometric Analysis (IGA) based on incremental methods for simulating the stamping process has been researched. To the best of our knowledge, however, few studies have combined IGA and One-step inverse approach which is based on total deformation theory of plasticity. A key step for One-step inverse IGA is to estimate a good initial solution. Traditional mesh-based initial solution algorithms for One-step inverse approach are not suitable for One-step inverse NURBS-based IGA. In this paper, we presented a method which can rapidly unfold the undevelopable NURBS surface onto a planar domain and obtain a good initial solution estimation for One-step inverse IGA. The key idea of the presented method is unfolding the control net of a NURBS surface for isogeometric analysis by energy-based initial solution estimation algorithm. In addition, we developed a “cutting-stitching” algorithm which can separate a complex control net into several parts with simple shapes. Numerical examples illustrate the initial solutions using the presented method are approaching the final results by One-step inverse finite element method. This implies that the iterative steps and computational time of One-step inverse IGA will be reduced significantly compared with that of One-step inverse finite element method. [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 SCRBE (Static-Condensation Reduced-Basis-Element) method is a component-to-system model order reduction approach for efficient many-query and real-time treatment of linear partial differential equations characterized by many spatially distributed constitutive, geometry, and topology parameters. In this paper we incorporate the SCRBE approach into a framework for analysis of problems in solid mechanics which are largely linear with the exception of local nonlinearities. In particular, we exploit a linear–nonlinear domain decomposition to develop a hybrid formulation: we consider a SCRBE approximation over the (assumed predominant) part of the domain associated to a linear elasticity model; we revert to a full finite element (FE) approximation over the part of the domain associated to the locally nonlinear model. We adapt the SCRBE port training procedures to anticipate the behavior of the field over the linear–nonlinear interface and hence ensure an accurate solution over the entire domain. We choose a globally conforming approximation which exploits the intrinsic port structure of the SCRBE method and yields a decoupled “non-invasive” formulation of the respective linear and nonlinear blocks of the residual vector and (Newton) Jacobian matrix. We present numerical results for several local nonlinearities – elastic contact with and without friction, plastic (contact) – which demonstrate substantial savings relative to standard FE approximation over the entire domain, in particular in the regime in which the linear part of the domain represents the majority of the (FE) degrees of freedom. [ABSTRACT FROM AUTHOR]