A FETI‐DP method for the parallel iterative solution of indefinite and complex‐valued solid and shell vibration problems

The dual‐primal finite element tearing and interconnecting (FETI‐DP) domain decomposition method (DDM) is extended to address the iterative solution of a class of indefinite problems of the form ( K ?σ² M ) u = f , and a class of complex problems of the form ( K ?σ² M +iσ D ) u = f , where K , M , and D are three real symmetric matrices arising from the finite element discretization of solid and shell dynamic problems, i is the imaginary complex number, and σ is a real positive number. A key component of this extension is a new coarse problem based on the free‐space solutions of Navier's equations of motion. These solutions are waves, and therefore the resulting DDM is reminiscent of the FETI‐H method. For this reason, it is named here the FETI‐DPH method. For a practically large σ range, FETI‐DPH is shown numerically to be scalable with respect to all of the problem size, substructure size, and number of substructures. The CPU performance of this iterative solver is illustrated on a 40‐processor computing system with the parallel solution, for various σ ranges, of several large‐scale, indefinite, or complex‐valued systems of equations associated with shifted eigenvalue and forced frequency response structural dynamics problems. Copyright © 2005 John Wiley & Sons, Ltd.     

Adaptive frequency windowing for multifrequency solutions in structural acoustics based on the matrix Padé‐via‐Lanczos algorithm

For many problems in structural acoustics, it is desired to obtain solutions at many frequencies over a large range in the frequency domain. A reduced‐order multifrequency algorithm based on matrix Padé approximation, using the matrix Padé‐via‐Lanczos (MPVL) connection, has been previously used to solve both exterior and interior acoustic problems. However, the method is not guaranteed to give the correct solution across the entire frequency region of interest, but only locally around a reference frequency. An adaptive frequency windowing scheme is introduced to address this shortcoming for practical application of this method. The application of this algorithm to tightly coupled problems in interior structural acoustics is presented. Copyright © 2007 John Wiley & Sons, Ltd.     

Robin‐Neumann transmission conditions for fluid‐structure coupling: Embedded boundary implementation and parameter analysis

Partitioned procedures are appealing for solving complex fluid‐structure interaction (FSI) problems, as they allow existing computational fluid dynamics (CFD) and computational structural dynamics algorithms and solvers to be combined and reused. However, for problems involving incompressible flow and strong added‐mass effect (eg, heavy fluid and slender structure), partitioned procedures suffer from numerical instability, which typically requires additional subiterations between the fluid and structural solvers, hence significantly increasing the computational cost. This paper investigates the use of Robin‐Neumann transmission conditions to mitigate the above instability issue. Firstly, an embedded Robin boundary method is presented in the context of projection‐based incompressible CFD and finite element–based computational structural dynamics. The method utilizes operator splitting and a modified ghost fluid method to enforce the Robin transmission condition on fluid‐structure interfaces embedded in structured non–body‐conforming CFD grids. The method is demonstrated and verified using the Turek and Hron benchmark problem, which involves a slender beam undergoing large transient deformation in an unsteady vortex‐dominated channel flow. Secondly, this paper investigates the effect of the combination parameter in the Robin transmission condition, ie, α_f, on numerical stability and solution accuracy. This paper presents a numerical study using the Turek and Hron benchmark problem and an analytical study using a simplified FSI model featuring an Euler‐Bernoulli beam interacting with a two‐dimensional incompressible inviscid flow. Both studies reveal a trade‐off between stability and accuracy: smaller values of α_f tend to improve numerical stability, yet deteriorate the accuracy of the partitioned solution. Using the simplified FSI model, the critical value of α_f that optimizes this trade‐off is derived and discussed.     

A novel non‐linearly explicit second‐order accurate L‐stable methodology for finite deformation: hypoelastic/hypoelasto‐plastic structural dynamics problems

A novel non‐linearly explicit second‐order accurate L‐stable computational methodology for integrating the non‐linear equations of motion without non‐linear iterations during each time step, and the underlying implementation procedure is described. Emphasis is placed on illustrative non‐linear structural dynamics problems employing both total/updated Lagrangian formulations to handle finite deformation hypoelasticity/hypoelasto‐plasticity models in conjunction with a new explicit exact integration procedure for a particular rate form constitutive equation. Illustrative numerical examples are shown to demonstrate the robustness of the overall developments for non‐linear structural dynamics applications. Copyright © 2004 John Wiley & Sons, Ltd.     

High‐Performance Microwave‐Derived Multi‐Principal Element Alloy Coatings for Tribological Application

The authors report the development of Al_xCoCrFeNi (x = 0.1 to 3) high entropy alloy (HEA) coatings using a simple and straightforward microwave technique. The microstructure of the developed coatings is composed of a cellular structure and diffused interface with the substrate. The microstructure of the HEA coatings varies as a direct function of Al content. An increase in Al fraction shows structural transformation from FCC to BCC along with the evolution of σ and B₂ as the major secondary phases. The diffusion of Mo from the substrate enhances the mixing entropy and promotes σ‐phase formation. The HEA coatings show significantly high hardness compared to SS316L substrate steel (227 HV) with a maximum value of 726 HV observed for three‐molar composition. The fracture toughness exhibits an inverse correlation with the Al fraction with the highest value of around 49 MPa m^1/2 observed for Al_0.1CoCrFeNi coating. The equimolar coating composition shows lowest erosion rates among all the tested samples due to optimum combination of the mechanical properties. The erosion resistance of the equimolar coating is 2 to 5 times higher than steel substrate and around 1.5 times higher than the non‐equimolar counterparts depending upon the impingement angles.

     

A finite element approach combining a reduced‐order system,Padé approximants,and an adaptive frequency windowing for fast multi‐frequency solution of poro‐acoustic problems

In this work, a solution strategy is investigated for the resolution of multi‐frequency structural‐acoustic problems including 3D modeling of poroelastic materials. The finite element method is used, together with a combination of a modal‐based reduction of the poroelastic domain and a Padé‐based reconstruction approach. It thus takes advantage of the reduced‐size of the problem while further improving the computational efficiency by limiting the number of frequency resolutions of the full‐sized problem. An adaptive procedure is proposed for the discretization of the frequency range into frequency intervals of reconstructed solution. The validation is presented on a 3D poro‐acoustic example. Copyright © 2013 John Wiley & Sons, Ltd.     

Higher‐order modal transformation for reduced‐order modeling of linear systems undergoing global parametric variations

Previously, a novel parametric reduced‐order model technique for linear systems was developed based on a frequency‐domain formulation and the so‐called modally equivalent perturbed system. The main advantage of the scheme is that it isolates all the perturbed matrices into a forcing term, allowing for a simple and powerful analysis based on the ordinary differential equation with the forcing input. It was shown that when the parameter variation is limited to a finite dimension, it yields exceptionally accurate reduced‐order models for a wide range of parameter values. In this paper, the original method is improved to cover a larger‐dimensional domain and the global domain of the variation by adding higher‐order terms in the formulation. It is shown that when expressed in powers of incremental matrices, the new formula resembles a well‐known series expansion. The improved parametric reduced‐order model is demonstrated for a computational fluid dynamics model of unsteady air flow around a two‐dimensional airfoil in subsonic flows with Mach variation.     

Efficient non‐linear model reduction via a least‐squares Petrov–Galerkin projection and compressive tensor approximations

A Petrov–Galerkin projection method is proposed for reducing the dimension of a discrete non‐linear static or dynamic computational model in view of enabling its processing in real time. The right reduced‐order basis is chosen to be invariant and is constructed using the Proper Orthogonal Decomposition method. The left reduced‐order basis is selected to minimize the two‐norm of the residual arising at each Newton iteration. Thus, this basis is iteration‐dependent, enables capturing of non‐linearities, and leads to the globally convergent Gauss–Newton method. To avoid the significant computational cost of assembling the reduced‐order operators, the residual and action of the Jacobian on the right reduced‐order basis are each approximated by the product of an invariant, large‐scale matrix, and an iteration‐dependent, smaller one. The invariant matrix is computed using a data compression procedure that meets proposed consistency requirements. The iteration‐dependent matrix is computed to enable the least‐squares reconstruction of some entries of the approximated quantities. The results obtained for the solution of a turbulent flow problem and several non‐linear structural dynamics problems highlight the merit of the proposed consistency requirements. They also demonstrate the potential of this method to significantly reduce the computational cost associated with high‐dimensional non‐linear models while retaining their accuracy. Copyright © 2010 John Wiley & Sons, Ltd.     

Reduced‐order modeling of parameterized PDEs using time–space‐parameter principal component analysis

This paper presents a methodology for constructing low‐order surrogate models of finite element/finite volume discrete solutions of parameterized steady‐state partial differential equations. The construction of proper orthogonal decomposition modes in both physical space and parameter space allows us to represent high‐dimensional discrete solutions using only a few coefficients. An incremental greedy approach is developed for efficiently tackling problems with high‐dimensional parameter spaces. For numerical experiments and validation, several non‐linear steady‐state convection–diffusion–reaction problems are considered: first in one spatial dimension with two parameters, and then in two spatial dimensions with two and five parameters. In the two‐dimensional spatial case with two parameters, it is shown that a 7 × 7 coefficient matrix is sufficient to accurately reproduce the expected solution, while in the five parameters problem, a 13 × 6 coefficient matrix is shown to reproduce the solution with sufficient accuracy. The proposed methodology is expected to find applications to parameter variation studies, uncertainty analysis, inverse problems and optimal design. Copyright © 2009 John Wiley & Sons, Ltd.     

An efficient indirect boundary element technique for multi‐frequency acoustic analysis

An efficient indirect boundary element solution procedure for the analysis of multi‐frequency acoustic problems is developed by incorporating techniques that improve the efficiency of the integration and matrix solution phases of the computing process. The integration phase is made efficient by computing the system matrices at few predetermined key frequencies only and then evaluating the matrices at other intermediate frequencies by quadratic interpolation. The matrix solution process is made efficient by iterating the solutions using the factored form of the key frequency matrices. The effectiveness of the present development is confirmed by solving a number of example problems. Copyright © 1999 John Wiley & Sons, Ltd.     

Inverse Sub‐Structuring Method for Multi‐Coordinate Coupled Product Transport System

A new high‐accuracy transfer function is selected, and an inverse sub‐structuring method is developed for the analysis of the dynamic characteristics of a three‐sub‐structure coupled product transport system. The closed‐form analytical solution to inverse sub‐structuring analysis of multi‐coordinate coupled multi‐ sub‐structure product transport system is derived. The proposed method is validated by a lumped mass spring damper model; the predicted frequency response functions (FRFs) of sub‐structures and the coupling stiffness, in addition to the most concerned system‐level FRF, are compared with the direct computations, showing exact agreement. Then, FRF tests of a physical prototype of the multi‐coordinate coupled product transport system with three sub‐structures are performed to further check the accuracy of the suggested method. The method developed offers an approach to predict the unknown sub‐structure‐level FRFs and coupling stiffness purely from system‐level FRFs. The suggested method may help obtain the main controlling factors and contributions from the various structure‐borne paths for the product transport system, which may certainly facilitate the cushioning packaging design. Copyright © 2013 John Wiley & Sons, Ltd.     

An improved continued‐fraction‐based high‐order transmitting boundary for time‐domain analyses in unbounded domains

A high‐order local transmitting boundary to model the propagation of acoustic or elastic, scalar or vector‐valued waves in unbounded domains of arbitrary geometry is proposed. It is based on an improved continued‐fraction solution of the dynamic stiffness matrix of an unbounded medium. The coefficient matrices of the continued‐fraction expansion are determined recursively from the scaled boundary finite element equation in dynamic stiffness. They are normalised using a matrix‐valued scaling factor, which is chosen such that the robustness of the numerical procedure is improved. The resulting continued‐fraction solution is suitable for systems with many DOFs. It converges over the whole frequency range with increasing order of expansion and leads to numerically more robust formulations in the frequency domain and time domain for arbitrarily high orders of approximation and large‐scale systems. Introducing auxiliary variables, the continued‐fraction solution is expressed as a system of linear equations in iω in the frequency domain. In the time domain, this corresponds to an equation of motion with symmetric, banded and frequency‐independent coefficient matrices. It can be coupled seamlessly with finite elements. Standard procedures in structural dynamics are directly applicable in the frequency and time domains. Analytical and numerical examples demonstrate the superiority of the proposed method to an existing approach and its suitability for time‐domain simulations of large‐scale systems. Copyright © 2011 John Wiley & Sons, Ltd.     

Review and assessment of interpolatory model order reduction methods for frequency response structural dynamics and acoustics problems

Frequency sweeps in structural dynamics, acoustics, and vibro‐acoustics require evaluating frequency response functions for a large number of frequencies. The brute force approach for performing these sweeps leads to the solution of a large number of large‐scale systems of equations. Several methods have been developed for alleviating this computational burden by approximating the frequency response functions. Among these, interpolatory model order reduction methods are perhaps the most successful. This paper reviews this family of approximation methods with particular attention to their applicability to specific classes of frequency response problems and their performance. It also includes novel aspects pertaining to the iterative solution of large‐scale systems of equations in the context of model order reduction and frequency sweeps. All reviewed computational methods are illustrated with realistic, large‐scale structural dynamic, acoustic, and vibro‐acoustic analyses in wide frequency bands. These highlight both the potential of these methods for reducing CPU time and their limitations. Copyright © 2012 John Wiley & Sons, Ltd.     

A time‐parallel implicit method for accelerating the solution of non‐linear structural dynamics problems

The parallel implicit time‐integration algorithm (PITA) is among a very limited number of time‐integrators that have been successfully applied to the time‐parallel solution of linear second‐order hyperbolic problems such as those encountered in structural dynamics. Time‐parallelism can be of paramount importance to fast computations, for example, when space‐parallelism is unfeasible as in problems with a relatively small number of degrees of freedom in general, and reduced‐order model applications in particular, or when reaching the fastest possible CPU time is desired and requires the exploitation of both space‐ and time‐parallelisms. This paper extends the previously developed PITA to the non‐linear case. It also demonstrates its application to the reduction of the time‐to‐solution on a Linux cluster of sample non‐linear structural dynamics problems. Copyright © 2008 John Wiley & Sons, Ltd.     

Two‐scale computational modelling of water flow in unsaturated soils containing irregular‐shaped inclusions

The focus of this paper is two‐dimensional computational modelling of water flow in unsaturated soils consisting of weakly conductive disconnected inclusions embedded in a highly conductive connected matrix. When the inclusions are small, a two‐scale Richards’ equation‐based model has been proposed in the literature taking the form of an equation with effective parameters governing the macroscopic flow coupled with a microscopic equation, defined at each point in the macroscopic domain, governing the flow in the inclusions. This paper is devoted to a number of advances in the numerical implementation of this model. Namely, by treating the micro‐scale as a two‐dimensional problem, our solution approach based on a control volume finite element method can be applied to irregular inclusion geometries, and, if necessary, modified to account for additional phenomena (e.g. imposing the macroscopic gradient on the micro‐scale via a linear approximation of the macroscopic variable along the microscopic boundary). This is achieved with the help of an exponential integrator for advancing the solution in time. This time integration method completely avoids generation of the Jacobian matrix of the system and hence eases the computation when solving the two‐scale model in a completely coupled manner. Numerical simulations are presented for a two‐dimensional infiltration problem. Copyright © 2014 John Wiley & Sons, Ltd.     

Exploiting semantics of temporal multi‐scale methods to optimize multi‐level mesh partitioning

Multi‐scale problems are often solved by decomposing the problem domain into multiple subdomains, solving them independently using different levels of spatial and temporal refinement, and coupling the subdomain solutions back to obtain the global solution. Most commonly, finite elements are used for spatial discretization, and finite difference time stepping is used for time integration. Given a finite element mesh for the global problem domain, the number of possible decompositions into subdomains and the possible choices for associated time steps is exponentially large, and the computational costs associated with different decompositions can vary by orders of magnitude. The problem of finding an optimal decomposition and the associated time discretization that minimizes computational costs while maintaining accuracy is nontrivial. Existing mesh partitioning tools, such as METIS, overlook the constraints posed by multi‐scale methods and lead to suboptimal partitions with a high performance penalty. We present a multi‐level mesh partitioning approach that exploits domain‐specific knowledge of multi‐scale methods to produce nearly optimal mesh partitions and associated time steps automatically. Results show that for multi‐scale problems, our approach produces decompositions that outperform those produced by state‐of‐the‐art partitioners like METIS and even those that are manually constructed by domain experts. Copyright © 2017 John Wiley & Sons, Ltd.     

Projection‐based model reduction for contact problems

To be feasible for computationally intensive applications such as parametric studies, optimization, and control design, large‐scale finite element analysis requires model order reduction. This is particularly true in nonlinear settings that tend to dramatically increase computational complexity. Although significant progress has been achieved in the development of computational approaches for the reduction of nonlinear computational mechanics models, addressing the issue of contact remains a major hurdle. To this effect, this paper introduces a projection‐based model reduction approach for both static and dynamic contact problems. It features the application of a non‐negative matrix factorization scheme to the construction of a positive reduced‐order basis for the contact forces, and a greedy sampling algorithm coupled with an error indicator for achieving robustness with respect to model parameter variations. The proposed approach is successfully demonstrated for the reduction of several two‐dimensional, simple, but representative contact and self contact computational models. Copyright © 2015 John Wiley & Sons, Ltd.     

Precorrected FFT accelerated BEM for large‐scale transient elastodynamic analysis using frequency‐domain approach

A precorrected fast Fourier transform (pFFT) accelerated boundary element method (BEM) for large‐scale transient elastodynamic analysis is developed and described in this paper. The frequency‐domain approach is used. To overcome the ‘wrap‐around’ problem associated with the discrete Fourier transform, the exponential window method (EWM) is employed and incorporated in the frequency‐domain BEM. An improved implementation scheme of the pFFT method based on polynomial interpolation technique is developed and applied to accelerate the elastodynamic BEM. This new scheme reduces the memory required to save the convolution matrix by a factor of 8. To further improve the efficiency of the code, a newly developed linear system solver based on the induced dimension reduction method is employed. Its performance is investigated and compared with that of the well‐known GMRES. The accuracy and computational efficiency of the method are evaluated and demonstrated by three examples: a classical benchmark, a plate subject to an impact loading and a porous cube with nearly half million DOFs subject to a step traction loading. Both analytical and experimental results are employed to validate the method. It has been found that the EWM can effectively resolve the wrap‐around problem and accurate time responses for an arbitrarily chosen time period can be obtained. Copyright © 2011 John Wiley & Sons, Ltd.