Interface‐reduction for the Craig–Bampton and Rubin method applied to FE–BE coupling with a large fluid–structure interface

Component mode‐based model‐order reduction (MOR) methods like the Craig–Bampton method or the Rubin method are known to be limited to structures with small coupling interfaces. This paper investigates two interface‐reduction methods for application of MOR to systems with large coupling interfaces: for the Craig–Bampton method a direct reduction method based on strain energy considerations is investigated. Additionally, for the Rubin method an iterative reduction scheme is proposed, which incrementally constructs the reduction basis. Hereby, attachment modes are tested if they sufficiently enlarge the spanned subspace of the current reduction basis. If so, the m‐orthogonal part is used to augment the basis. The methods are applied to FE–BE coupled systems in order to predict the vibro‐acoustic behavior of structures, which are partly immersed in water. Hereby, a strong coupling scheme is employed, since for dense fluids the feedback of the acoustic pressure onto the structure is not negligible. For two example structures, the efficiency of the reduction methods with respect to numerical effort, memory consumption and computation time is compared with the exact full‐order solution. Copyright © 2008 John Wiley & Sons, Ltd.     

Fast exact linear and non‐linear structural reanalysis and the Sherman–Morrison–Woodbury formulas

Several exact fast static structural reanalysis techniques, introduced by researchers mostly for truss structures and some for frames and plate structures, are reviewed. Most utilize the property that the solution of a system of linear equations can be updated inexpensively when the matrix is changed by a low‐rank increment. This paper shows that these methods are variants of the well‐known Sherman–Morrison and Woodbury (SMW) formulas for the update of the inverse of a matrix. In addition, the paper extends the low‐cost linear reanalysis in the spirit of the SMW formulas to some non‐linear reanalysis problems. For a linear reanalysis, the extension reduces to the SMW formulas. Copyright © 2001 John Wiley & Sons, Ltd.     

A preconditioned conjugate gradient approach to structural reanalysis for general layout modifications

This paper presents a preconditioned conjugate gradient approach to structural static reanalysis for general layout modifications. It is suitable for all types of layout modifications, including the general case in which some original members and nodes are deleted and other new members and nodes are added concurrently. The approach is based on the preconditioned conjugate gradient technique. The preconditioner is constructed, and an efficient implementation for applying the preconditioner is presented, which requires the factorization of the stiffness matrix corresponding to the newly added degrees of freedom only. In particular, the approach can adaptively monitor the accuracy of approximate solutions. Numerical examples show that the condition number of the preconditioned matrix is remarkably reduced. Therefore, the fast convergence and accurate results can be achieved by the approach. Copyright © 2006 John Wiley & Sons, Ltd.     

Coupling of non‐conforming meshes in a component mode synthesis method

A common mesh refinement‐based coupling technique is embedded into a component mode synthesis method, Craig–Bampton. More specifically, a common mesh is generated between the non‐conforming interfaces of the coupled structures, and the compatibility constraints are enforced on that mesh via L2‐minimization. This new integrated method is suitable for structural dynamic analysis problems where the substructures may have non‐conforming curvilinear and/or surface interface meshes. That is, coupled substructures may have different element types such as shell, solid, and/or beam elements. The proposed method is implemented into a commercial finite element software, B2000++, and its demonstration is carried out using an academic and industry oriented test problems. Copyright © 2013 John Wiley & Sons, Ltd.     

A method for modal reanalysis of topological modifications of structures

A method for structural modal reanalysis for three cases of topological modifications, the number of degrees of freedom (DOFs) is unchanged, decreased, and increased, is presented. In this method, the newly added DOFs are linked to the original DOFs of the modified structure by means of the dynamic reduction so as to obtain the condensed equation. The methods for forming the stiffness and mass increments, Δ K and Δ M , resulting from the three cases of topological modifications of structures are discussed. The extended Kirsch method is used to improve the accuracy of the starting solutions of the initial structure. And then, the eigenvectors of newly added DOFs resulting from topological modification can be recovered. At last, the Rayleigh–Ritz analysis is used to evaluate the eigenvalues and eigenvectors for the modified structure. Three numerical examples are given to illustrate the applications of the present approach. The results show that the proposed method is effective for structural modal reanalysis of three cases of the topological modifications and it is easy to implement on a computer. By comparing with previous method, it can be seen that the present method can give good approximate eigenvalues and eigenvectors, even if the topological modifications are very large. Copyright © 2005 John Wiley & Sons, Ltd.     

An efficient block preconditioner for Jacobian‐free global–local multiscale methods

In this paper, we develop a block preconditioner for Jacobian‐free global–local multiscale methods, in which the explicit computation of the Jacobian may be circumvented at the macroscale by using a Newton–Krylov process. Effective preconditioning is necessary for the Krylov subspace iterations (e.g. GMRES) to enhance computational efficiency. This is, however, challenging since no explicit information regarding the Jacobian matrix is available. The block preconditioning technique developed in this paper circumvents this problem by effectively deflating the spectrum of the Jacobian matrix at the current Newton step using information about only the Krylov subspaces corresponding to the Jacobian matrices in the previous Newton steps and their representations on those subspaces. This approach is optimal and results in exponential convergence of the GMRES iterations within each Newton step, thus minimizing expensive microscale computations without requiring explicit Jacobian formation in any step. In terms of both computational cost and storage requirements, the action of a single block of the preconditioner per GMRES step scales linearly as the number of degrees of freedom of the macroscale problem as well as the dimension of the invariant subspace of the preconditioned Jacobian matrix. Copyright © 2011 John Wiley & Sons, Ltd.     

Optimal component mode synthesis for medium frequency problem

The component mode synthesis (CMS) with fixed interface (denoted Craig–Bampton) method uses a combination of static and dynamic modes. The usual definition of this CMS leads to a coupling between static and dynamic modes which are not orthogonal with respect to the stiffness matrix. This type of basis is not well suited for dynamic explicit computations, because the resulting mass matrix is not diagonal. If one keeps the same basis mode set but uses an orthogonalization process with respect to the mass matrix, the quality of the reduced Craig–Bampton system is kept but the basis vectors are combined differently. The aim of this paper is to propose a new way to control the accuracy of the reduced dynamic system for a specific frequency domain. Thus a new CMS is defined in order to be accurate in the medium frequency range. Copyright © 2010 John Wiley & Sons, Ltd.     

ANM for stationary Navier–Stokes equations and with Petrov–Galerkin formulation

This paper deals with the use of the asymptotic numerical method (ANM) for solving non‐linear problems, with particular emphasis on the stationary Navier–Stokes equation and the Petrov–Galerkin formulation. ANM is a combination of a perturbation technique and a finite element method allowing to transform a non‐linear problem into a succession of linear ones that admit the same tangent matrix. This method has been applied with success in non‐linear elasticity and fluid mechanics. In this paper, we apply the same kind of technique for solving Navier–Stokes equation with the so‐called Petrov–Galerkin weighting. The main difficulty comes from the fact that the non‐linearity is no more quadratic and it is not evident, in this case, to be able to compute a large number of terms of the perturbation series. Several examples of fluid mechanic are presented to demonstrate the performance of such a method. Copyright © 2001 John Wiley & Sons, Ltd.     

An efficient and high‐order frequency‐domain approach for transient acoustic–structure interactions in three dimensions

This paper is concerned with numerical solution of the transient acoustic–structure interaction problems in three dimensions. An efficient and higher‐order method is proposed with a combination of the exponential window technique and a fast and accurate boundary integral equation solver in the frequency‐domain. The exponential window applied to the acoustic–structure system yields an artificial damping to the system, which eliminates the wrap‐around errors brought by the discrete Fourier transform. The frequency‐domain boundary integral equation approach relies on accurate evaluations of relevant singular integrals and fast computation of nonsingular integrals via the method of equivalent source representations and the fast Fourier transform. Numerical studies are presented to demonstrate the accuracy and efficiency of the method. Copyright © 2016 John Wiley & Sons, Ltd.     

Improved radial basis function fluid–structure coupling via efficient localized implementation

Previous work by the authors has developed a universal interpolation scheme, using radial basis functions (RBFs), which results in a unified formulation for robust fluid–structure interpolation and high‐quality mesh motion. The method has several significant advantages. Primarily, all volume mesh, structural mesh, and flow‐solver‐type dependence is removed entirely, as all operations are performed on totally arbitrary point clouds of any form. Hence, all connectivity requirements are removed from both the coupling and mesh motion problems. Furthermore, only matrix‐vector multiplications are required during unsteady simulation because dependence relations are computed once prior to any simulation and then remain constant. This property means that the method is both perfectly parallel and totally independent from the flow‐solver. However, the full method is expensive, since the dependence matrix between two sets of points is N × N. The fluid–structure coupling behaviour can also be influenced by parameters used in the interpolation. To alleviate these difficulties a more efficient form of the RBF fluid–structure coupling is presented, which also greatly reduces the interpolation parameter influence. A pointwise form of the partition of unity approach is developed that localizes the interpolation, with results presented for static aeroelastic simulations of the Brite‐Euram multi‐disciplinary optimization wing using a very fine mesh containing 58 000 surface points. It is shown that a 58 × reduction in data size is achieved, and equally importantly the interpolation has a much smaller influence on final aeroelastic results. Copyright © 2009 John Wiley & Sons, Ltd.     

An efficient coarse‐grained parallel algorithm for global–local multiscale computations on massively parallel systems

The existing global–local multiscale computational methods, using finite element discretization at both the macro‐scale and micro‐scale, are intensive both in terms of computational time and memory requirements and their parallelization using domain decomposition methods incur substantial communication overhead, limiting their application. We are interested in a class of explicit global–local multiscale methods whose architecture significantly reduces this communication overhead on massively parallel machines. However, a naïve task decomposition based on distributing individual macro‐scale integration points to a single group of processors is not optimal and leads to communication overheads and idling of processors. To overcome this problem, we have developed a novel coarse‐grained parallel algorithm in which groups of macro‐scale integration points are distributed to a layer of processors. Each processor in this layer communicates locally with a group of processors that are responsible for the micro‐scale computations. The overlapping groups of processors are shown to achieve optimal concurrency at significantly reduced communication overhead. Several example problems are presented to demonstrate the efficiency of the proposed algorithm. Copyright © 2009 John Wiley & Sons, Ltd.     

An efficient method of solving the Navier–Stokes equations for flow control

A new method of solving the Navier–Stokes equations efficiently by reducing their number of modes is proposed in the present paper. It is based on the Karhunen–Loève decomposition which is a technique of obtaining empirical eigenfunctions from the experimental or numerical data of a system. Employing these empirical eigenfunctions as basis functions of a Galerkin procedure, one can a priori limit the function space considered to the smallest linear subspace that is sufficient to describe the observed phenomena, and consequently reduce the Navier–Stokes equation defined on a complicated geometry to a set of ordinary differential equations with a minimum degree of freedom. The present algorithm is well suited for the problems of flow control or optimization, where one has to compute the flow field repeatedly using the Navier–Stokes equation but one can also estimate the approximate solution space of the flow field based on the range of control variables. The low-dimensional dynamic model of viscous fluid flow derived by the present method is shown to produce accurate flow fields at a drastically reduced computational cost when compared with the finite difference solution of the Navier–Stokes equation. © 1998 John Wiley & Sons, Ltd.     

An alternative TLM method for steady‐state convection–diffusion

Recent papers have introduced a novel and efficient scheme, based on the transmission line modelling (TLM) method, for solving one‐dimensional steady‐state convection–diffusion problems. This paper introduces an alternative method. It presents results obtained using both techniques, which suggest that the new scheme outlined in this paper is the more accurate and efficient of the two. Copyright © 2009 John Wiley & Sons, Ltd.     

Reliability–sensitivity analysis using dimension reduction methods and saddlepoint approximations

Reliability–sensitivity, which is considered as an essential component in engineering design under uncertainty, is often of critical importance toward understanding the physical systems underlying failure and modifying the design to mitigate and manage risk. This paper presents a new computational tool for predicting reliability (failure probability) and reliability–sensitivity of mechanical or structural systems subject to random uncertainties in loads, material properties, and geometry. The dimension reduction method is applied to compute response moments and their sensitivities with respect to the distribution parameters (e.g., shape and scale parameters, mean, and standard deviation) of basic random variables. Saddlepoint approximations with truncated cumulant generating functions are employed to estimate failure probability, probability density functions, and cumulative distribution functions. The rigorous analytic derivation of the parameter sensitivities of the failure probability with respect to the distribution parameters of basic random variables is derived. Results of six numerical examples involving hypothetical mathematical functions and solid mechanics problems indicate that the proposed approach provides accurate, convergent, and computationally efficient estimates of the failure probability and reliability–sensitivity. Copyright © 2012 John Wiley & Sons, Ltd.     

The enriched space–time finite element method (EST) for simultaneous solution of fluid–structure interaction

The paper introduces a weighted residual‐based approach for the numerical investigation of the interaction of fluid flow and thin flexible structures. The presented method enables one to treat strongly coupled systems involving large structural motion and deformation of multiple‐flow‐immersed solid objects. The fluid flow is described by the incompressible Navier–Stokes equations. The current configuration of the thin structure of linear elastic material with non‐linear kinematics is mapped to the flow using the zero iso‐contour of an updated level set function. The formulation of fluid, structure and coupling conditions uniformly uses velocities as unknowns. The integration of the weak form is performed on a space–time finite element discretization of the domain. Interfacial constraints of the multi‐field problem are ensured by distributed Lagrange multipliers. The proposed formulation and discretization techniques lead to a monolithic algebraic system, well suited for strongly coupled fluid–structure systems. Embedding a thin structure into a flow results in non‐smooth fields for the fluid. Based on the concept of the extended finite element method, the space–time approximations of fluid pressure and velocity are properly enriched to capture weakly and strongly discontinuous solutions. This leads to the present enriched space–time (EST) method. Numerical examples of fluid–structure interaction show the eligibility of the developed numerical approach in order to describe the behavior of such coupled systems. The test cases demonstrate the application of the proposed technique to problems where mesh moving strategies often fail. Copyright © 2007 John Wiley & Sons, Ltd.     

On the validity of Hall–Petch equation for single-phase β Ti–Al–Nb alloys undergoing stress-induced martensitic transformation

The yield strengths of Ti–Al–Nb alloys, which undergo stress-induced martensitic transformation, prior to the onset of plastic deformation during tensile testing, were found to obey the Hall–Petch relationship with grain size. The overall friction stress was observed to decrease with increase in Nb content while it remained more or less unchanged with increase in Al content in these alloys. On the other hand, the overall , the unpinning constant, which is an index of the efficiency of boundaries as obstacle to dislocation motion, was found to increase with increase in Nb and decrease with increase in Al content in these alloys.     

Topology optimization for stationary fluid–structure interaction problems using a new monolithic formulation

This paper outlines a new procedure for topology optimization in the steady‐state fluid–structure interaction (FSI) problem. A review of current topology optimization methods highlights the difficulties in alternating between the two distinct sets of governing equations for fluid and structure dynamics (hereafter, the fluid and structural equations, respectively) and in imposing coupling boundary conditions between the separated fluid and solid domains. To overcome these difficulties, we propose an alternative monolithic procedure employing a unified domain rather than separated domains, which is not computationally efficient. In the proposed analysis procedure, the spatial differential operator of the fluid and structural equations for a deformed configuration is transformed into that for an undeformed configuration with the help of the deformation gradient tensor. For the coupling boundary conditions, the divergence of the pressure and the Darcy damping force are inserted into the solid and fluid equations, respectively. The proposed method is validated in several benchmark analysis problems. Topology optimization in the FSI problem is then made possible by interpolating Young's modulus, the fluid pressure of the modified solid equation, and the inverse permeability from the damping force with respect to the design variables. Copyright © 2009 John Wiley & Sons, Ltd.     

Evaluation by extended Hückel method on the hardness of the B–C–N materials

Hard materials, e.g. diamond and cubic boron nitride (c-BN), are widely applied to improve the lifetime and the performance of many kinds of cutting and forming tools. These materials are usually used at high temperature, so the study of stability on these materials at high temperature is very important. However, diamond is a low resistance to the oxidation, it should be replaced with the boron-based hard materials. Recently, boron–carbon–nitrogen (B–C–N) ternary materials are expected to possess a high hardness, a high thermal stability at high temperature. We estimated the hardness and the stability of B–C–N materials at high temperature by the extended Hückel method. The extended Hückel method is one of the molecular orbital calculations and needs the cluster model of materials for the calculation. The cluster model of B–C–N materials was regarded as a zinc blende structure. In the present work, we used two physical quantities, i.e. a cohesive energy and an energy fluctuation, as a measure of hardness and stability of materials. The cohesive energy indicates the coherence of bonds between atoms. The energy fluctuation shows the reactivity of materials. Hardness, structure, solid-state properties and reactivity of materials can be estimated from these physical quantities. When the composition of B–C–N materials was boron: 25 at.%, nitrogen: 25 at.% and carbon: 50 at.%, the cohesive energy was the lowest. This result implies B–C–N ternary materials are not harder than c-BN and/or diamond. Cubic-BN was the lowest energy fluctuation of B–C–N materials, and the energy fluctuation increased as increasing of carbon atom. The reactivity of B–C–N materials was high at a high temperature with an increase of carbon atoms. These results imply that B–C–N materials are not suitable for the hard cutting materials.     

A TLM method for steady‐state convection–diffusion: Some additions and refinements

A recent paper introduced a novel and efficient scheme, based on the transmission line modelling (TLM) method, for solving steady‐state convection–diffusion problems. This paper shows how this one‐dimensional scheme can be adapted to include reaction and source terms and how it can be implemented with non‐equidistant nodes. It introduces new ways of calculating the necessary model parameters which can improve the accuracy of the scheme, shows how steady‐state solutions can be obtained directly, and compares results with those from two finite difference (FD) methods. While the cost of implementation is higher than for the FD schemes, the new TLM scheme can be significantly more accurate, especially when convection dominates. Copyright © 2008 John Wiley & Sons, Ltd.