Local proper generalized decomposition

One of the main difficulties that a reduced‐order method could face is the poor separability of the solution. This problem is common to both a posteriori model order reduction (proper orthogonal decomposition, reduced basis) and a priori [proper generalized decomposition (PGD)] model order reduction. Early approaches to solve it include the construction of local reduced‐order models in the framework of POD. We present here an extension of local models in a PGD—and thus, a priori—context. Three different strategies are introduced to estimate the size of the different patches or regions in the solution manifold where PGD is applied. As will be noticed, no gluing or special technique is needed to deal with the resulting set of local reduced‐order models, in contrast to most proper orthogonal decomposition local approximations. The resulting method can be seen as a sort of a priori manifold learning or nonlinear dimensionality reduction technique. Examples are shown that demonstrate pros and cons of each strategy for different problems.     

A wavelet–Galerkin scheme for analysis of large‐scale problems on simple domains

A wavelet–Galerkin scheme tailored to address the numerical solution of large‐scale boundary value problems defined on domains of simple geometry is presented. The variation of parameters, e.g. material properties, within the domain is arbitrary but the method is specifically designed to solve problems where parameters vary in raster‐like fashion. Boundary conditions are imposed via Lagrange multipliers using a fictitious domain approach. A preconditioner specially designed for this problem is developed to guarantee that convergence of conjugate gradient algorithms is quick and insensitive to problem size. The strategy is applied to the solution of steady state, heat conduction problems in 2‐D, but it can be generalized without conceptual changes to 3‐D problems and to problems in linear elasticity. Copyright © 1999 John Wiley & Sons, Ltd.     

Real‐time direct integration of reduced solid dynamics equations

A new method is developed here for the real‐time integration of the equations of solid dynamics based on the use of proper orthogonal decomposition (POD)–proper generalized decomposition (PGD) approaches and direct time integration. The method is based upon the formulation of solid dynamics equations as a parametric problem, depending on their initial conditions. A sort of black‐box integrator that takes the resulting displacement field of the current time step as input and (via POD) provides the result for the subsequent time step at feedback rates on the order of 1 kHz is obtained. To avoid the so‐called curse of dimensionality produced by the large amount of parameters in the formulation (one per degree of freedom of the full model), a combined POD–PGD strategy is implemented. Examples that show the promising results of this technique are included. Copyright © 2014 John Wiley & Sons, Ltd.     

Young's modulus of porous brittle solids

A new equationE =E ₀ (1 –aP) ⁿ whereE andE ₀ are the Young's moduli at porosity,P, and zero, respectively, a andn are material constants, has been derived semi-empirically for describing the porosity dependence of Young's modulus of brittle solids. The equation satisfies quite well the exact theoretical solution for the values of Young's moduli at different porosities for model systems with ideal and non-ideal packing geometry. The equation shows excellent agreement with the data On- and-alumina over a wide range of porosity. Unlike the existing porosity-elastic modulus equations, the proposed equation satisfies the boundary conditions and is inherently capable of treating isometric closed pores as well as non-isometric interconnected pores. The parameters a and n provide information about the packing geometry and pore structure of the material.     

Extended stochastic FEM for diffusion problems with uncertain material interfaces

This paper is concerned with the prediction of heat transfer in composite materials with uncertain inclusion geometry. To numerically solve the governing equation, which is defined on a random domain, an approach based on the combination of the Extended finite element method (X-FEM) and the spectral stochastic finite element method is studied. Two challenges of the extended stochastic finite element method (X-SFEM) are choosing an enrichment function and numerical integration over the probability domain. An enrichment function, which is based on knowledge of the interface location, captures the C ⁰-continuous solution in the spatial and probability domains without a conforming mesh. Standard enrichment functions and enrichment functions tailored to X-SFEM are analyzed and compared, and the basic elements of a successful enrichment function are identified. We introduce a partition approach for accurate integration over the probability domain. The X-FEM solution is studied as a function of the parameters describing the inclusion geometry and the different enrichment functions. The efficiency and accuracy of a spectral polynomial chaos expansion and a finite element approximation in the probability domain are compared. Numerical examples of a two-dimensional heat conduction problem with a random inclusion show the spectral PC approximation with a suitable choice of enrichment function is as accurate and more efficient than the finite element approach. Though focused on heat transfer in composite materials, the techniques and observations in this paper are also applicable to other types of problems with uncertain geometry.     

Generalized debye series for acoustic scattering from objects of separable geometric shape

Summary In a classical paper of 1908, Debye has resolved the electromagnetic field scattered by a dielectric cylinder into a series of waves multiply internally reflected in the cylinder. For acoustic scattering by elastic cylinders, a corresponding series was derived from the conventional solution (obtained by satisfying the overall or global continuity conditions) by Brill and Überall, taking into account mode conversions of longitudinal (L) into transverse (T, shear) waves, or vice versa, upon internal scattering in a some-what involved fashion. In a series of papers, Gérard has shown that this approach could be greatly simplified by introducing local reflection and transmission coefficients at each interface, which is suitable for generalizing the Debye series to the case of elastic waves coupled by the continuity conditions at the external and each of any possible (multiple) internal interfaces of the scattering object. The approach is then applicable to all elastic objects for which surface and interfaces form coordinate surfaces of any separable geometry; the corresponding derivation is given here in the most general fashion, and is concretely illustrated by the examples of an elastic plate, infinite cylinder and sphere.     

Transient solutions to nonlinear acousto‐magneto‐mechanical coupling for axisymmetric MRI scanner design

In this work, we simulate the coupled physics describing a magnetic resonance imaging (MRI) scanner by using a higher‐order finite element discretisation and a Newton‐Raphson algorithm. To apply the latter, a linearisation of the nonlinear system of equations is necessary, and we consider two alternative approaches. In the first approach, ie, the nonlinear approach, there is no approximation from a physical standpoint, and the linearisation is performed about the current solution. In the second approach, ie, the linearised approach, we realise that the MRI problem can be described by small dynamic fluctuations about a dominant static solution and linearise about the latter. The linearised approach permits solutions in the frequency domain and provides a computationally efficient way to solve this challenging problem, as it allows the tangent stiffness matrix to be inverted independently of time or frequency. We focus on transient solutions to the coupled system of equations and address the following two important questions: (i) how good is the agreement between the computationally efficient linearised approach compared with the intensive nonlinear approach and (ii) over what range of MRI operating conditions can the linearised approach be expected to provide acceptable results for outputs of interest in an industrial context for MRI scanner design? We include a set of academic and industrially relevant examples to benchmark and illustrate our approach.     

The fourth-dimention concept in the finite element analysis of transient heat transfer problems

This study presents the finite element analysis for the transient heat transfer problems by introducing the Fourth-dimension concept. Time is treated as an additional dimension in the solution domain thereby Increasing the number of dimensions by one. For instance, a three-dimensional transient problem can be considered as a four-dimensional problem in the x, y, z, t domain and a two-dimensional transient problem can be considered as a three-dimensional steady-state problem in the x, y, t domain, respectively. The variational principle of the finite element method and the techniques existing for steady-state problems can be directly utilized. Numerical calculations were performed for heat conduction problems, laminar-turbulent convective heat transfer problems, and radiative heat transfer problems.     

Unilateral contact problems with fractal geometry and fractal friction laws: methods of calculation

The present paper deals with two interrelated subjects: the fractal geometry and the fractal behaviour in unilateral contact problems. More specifically, throughout this paper both the interfaces and the friction laws holding on these interfaces are modelled by means of the fractal geometry. It is important to notice here that the fractality of the induced friction laws takes into account the randomness of the interface asperities causing the friction forces. According to the fractal model introduced in this paper, both the fractal law and the fractal interface are considered to be graphs of two different fractal interpolation functions which are the “fixed points” of two contractive operators. Using this method, the fractal friction law is approximated by a sequence of nonmonotone possibly multivalued classical C ⁰-curves. The numerical treatment of each arizing nonmonotone problem is accomplished by an advanced solution method which approximates the nonmonotone problem by a sequence of monotone subproblems. Numerical applications from the static analysis of cracked structures with a prescribed fractal geometry and fractal interface laws are included in order to illustrate the theory.     

A parallel and multiscale strategy for the parametric study of transient dynamic problems with friction

The objective of this work is to develop an efficient strategy for the parametric study of dynamic problems involving contacts with friction. Our approach is based on the multiscale LATIN method with domain decomposition. This is a mixed method that deals with the forces and velocities at the interfaces between the different subdomains simultaneously. We propose to take advantage of the capability of the multiscale LATIN method, called the multiparametric strategy, to reuse the solution of a given problem in order to solve similar problems. This strategy has already been applied successfully to a variety of static problems; here, it is extended to dynamics. First, we present the multiscale strategy in dynamics. Then, we show how the multiparametric strategy can be extended to dynamics. We illustrate the capabilities of the method through an academic 3D example and the simulation of a bolted joint. Copyright © 2011 John Wiley & Sons, Ltd.     

Assembly of nanoscale building blocks at solution/solid interfaces

A chemical strategy has been purposely designed to hierarchically assemble nanoscale building blocks at the interface of solution/solid. Typically, a solution containing precursor of one component and a metal foil as metal source of another component were employed, on the basis of proposed chemical reactions on expected interfaces. Proper reaction parameters including temperature, pH value etc. were selected to adapt both chemical reactions in solution and on the metal surface. Consequently, at the interface of solution and metal foil, two kinds of nanoscale building blocks deposited simultaneously leading to the current hierarchical assembly. This strategy has been applied to the fabrication of a series of functional materials, including Nb₂O₅/TiO₂, Nb₂O₅/LiF and ZnO/Co₃O₄. The current strategy provides a convenient one-step route to achieve complex functional structures, which may have potential applications in a variety of fields such as solar cells, Li-ion batteries, electrochemical supercapacitors, catalysts as well as chemical, gas, and bio-sensors.     

Numerical analysis of effect of material and processing parameters on thickness distribution during superplastic die bulging of cavity sensitive materials

Abstract

The superplastic bulging of circular sheets clamped against axisymmetrical cylindrical dies has been analysed numerically by means of a rigid–viscoplastic finite element method, in which four node quadrilateral isoparametric elements are used with a Newton–Raphson non­linear solution scheme. Both effects of normal anisotropy and strain hardening in the material are considered and a modified Coulomb friction law is adopted. At the same time, the yield criterion suited for the superplastic forming process and the cavity damage evolution model deduced from continuum damage mechanics are applied to a finite element formulation. The influences of material parameters (the strain rate sensitivity exponent m, the strain hardening exponent n, the coefficient of normal anisotropy R) and processing parameters (pressure cycle, lubrication condition, die geometry) on the inhomogeneity of the thickness distribution are studied and discussed. A selection of the simulated results is compared with the experimental results, with good agreement.     

eXtended Stochastic Finite Element Method for the numerical simulation of heterogeneous materials with random material interfaces

An eXtended Stochastic Finite Element Method has been recently proposed for the numerical solution of partial differential equations defined on random domains. This method is based on a marriage between the eXtended Finite Element Method and spectral stochastic methods. In this article, we propose an extension of this method for the numerical simulation of random multi‐phased materials. The random geometry of material interfaces is described implicitly by using random level set functions. A fixed deterministic finite element mesh, which is not conforming to the random interfaces, is then introduced in order to approximate the geometry and the solution. Classical spectral stochastic finite element approximation spaces are not able to capture the irregularities of the solution field with respect to spatial and stochastic variables, which leads to a deterioration of the accuracy and convergence properties of the approximate solution. In order to recover optimal convergence properties of the approximation, we propose an extension of the partition of unity method to the spectral stochastic framework. This technique allows the enrichment of approximation spaces with suitable functions based on an a priori knowledge of the irregularities in the solution. Numerical examples illustrate the efficiency of the proposed method and demonstrate the relevance of the enrichment procedure. Copyright © 2010 John Wiley & Sons, Ltd.     

Replica time integrators

This paper is concerned with the classical problem of wave propagation in discrete models of nonuniform spatial resolution. We develop a new class of Replica Time Integrators (RTIs) that permit the two‐way transmission of thermal phonons across mesh interfaces. This two‐way transmissibility is accomplished by representing the state of the coarse regions by means of replica ensembles, consisting of collections of identical copies of the coarse regions. In dimension d, RTIs afford an O(n^d) speed‐up factor in sequential mode, and O(n^{d + 1}) in parallel, over regions that are coarsened n‐fold. In this work, we restrict ourselves to the solution of the 3d continuous wave equation, for both linear and non‐linear materials. By a combination of phase‐error analysis and numerical testing, we show that RTIs are convergent and result in exact two‐way transmissibility at the Courant–Friedrichs–Lewy limit for any angle of incidence. In this limit, RTIs allow step waves and high‐frequency harmonics to cross mesh interfaces in both directions without internal reflections or appreciable loss or addition of energy. The possible connections of RTIs with discrete‐to‐continuum approaches and, in particular, with the transition between molecular dynamics and continuum thermodynamics are also pointed to by way of future outlook. Copyright © 2011 John Wiley & Sons, Ltd.     

Fatigue crack propagation along interfaces of selective laser melting steel hybrid parts

Selective laser melting (SLM) is an emerging additive manufacturing technology, capable of producing complex geometry components. The current work studied both the effect of substrate material and mean stress on the fatigue crack growth behaviour along interfaces of bi‐material specimens, substrate, and part by SLM. Fatigue tests were carried out in agreement with ASTM E647 standard, using 6‐mm‐thick compact specimens. The substrate steel has only a negligible effect both on the fatigue crack propagation rate and on the crack path. The failure occurs in the material additively manufactured by SLM, near the interface. The mean stress produced only a reduced influence on the fatigue crack propagation rate in the Paris regime. For larger values of ΔK, where K_max approaches K_Ic, a significant influence of the mean stress was observed. In spite of nondetection of crack closure, the application of overloads promoted significant fatigue crack retardation, quite similar for both substrate materials, probably due to the crack bifurcation during the overload.     

Analytical investigation and parametric study of lateral impact behavior of pressurized pipelines and influence of internal pressure

This article provides a combined computational and analytical study to investigate the lateral impact behavior of pressurized pipelines and inspect all the parameters such as the outside diameter and internal pressure and evaluate how they affect such behavior. In this study, quartic polynomial functions are applied to formulate the maximum crushing force (F_max), maximum permanent displacement (W), and absorbed energy (E) of the pressurized pipelines during the impact problem. The effects of the diameter and pressure on F_max, W, and E are therefore illustrated through analyzing these functions. Response surfaces are also plotted based on the generated quartic polynomial functions and the quality (accuracy) of these functions are verified through several techniques.