Model reduction for dynamical systems with quadratic output

Finite element models for structures and vibrations often lead to second order dynamical systems with large sparse matrices. For large‐scale finite element models, the computation of the frequency response function and the structural response to dynamic loads may present a considerable computational cost. Padé via Krylov methods are widely used and are appreciated projection‐based model reduction techniques for linear dynamical systems with linear output. This paper extends the framework of the Krylov methods to systems with a quadratic output arising in linear quadratic optimal control or random vibration problems. Three different two‐sided model reduction approaches are formulated based on the Krylov methods. For all methods, the control (or right) Krylov space is the same. The difference between the approaches lies, thus, in the choice of the observation (or left) Krylov space. The algorithms and theory are developed for the particularly important case of structural damping. We also give numerical examples for large‐scale systems corresponding to the forced vibration of a simply supported plate and of an existing footbridge. In this case, a block form of the Padé via Krylov method is used. Copyright © 2012 John Wiley & Sons, Ltd.     

Matrix‐Padé via Lanczos solutions for vibrations of fluid–structure interaction

For multiple‐frequency full‐field solutions of the boundary value problem describing small fluid–structure interaction vibration superimposed on a nominal state with prestress, we propose an efficient reduced order method by constructing the full‐field matrix‐Padé approximant of its finite element matrix function. Exploiting the matrix‐Padé via Lanczos connection, the Padé coefficients are computed in a stable and efficient way via an unsymmetric, banded Lanczos process. The full‐field Padé‐type approximant is the result of one‐sided projection onto Krylov subspace, we established its order of accuracy, which is not maximal. The superiority of this method in terms of various problem dimensions and parameters is established by complexity analysis via flop counts. Numerical examples obtained by using a model problem verified the accuracy of this full‐field matrix‐Padé approximant. Copyright © 2010 John Wiley & Sons, Ltd.     

Polynomial chaos‐based extended Padé expansion in structural dynamics

The response of a random dynamical system is totally characterized by its probability density function (pdf). However, determining a pdf by a direct approach requires a high numerical cost; similarly, surrogate models such as direct polynomial chaos expansions are not generally efficient, especially around the eigenfrequencies of the dynamical system. In the present study, a new approach based on Padé approximants to obtain moments and pdf of the dynamic response in the frequency domain is proposed. A key difference between the direct polynomial chaos representation and the Padé representation is that the Padé approach has polynomials in both numerator and denominator. For frequency response functions, the denominator plays a vital role as it contains the information related to resonance frequencies, which are uncertain. A Galerkin approach in conjunction with polynomial chaos is proposed for the Padé approximation. Another physics‐based approach, utilizing polynomial chaos expansions of the random eigenmodes, is proposed and compared with the proposed Padé approach. It is shown that both methods give accurate results even if a very low degree of the polynomial expansion is used. The methods are demonstrated for two degree‐of‐freedom system with one and two uncertain parameters. Copyright © 2016 John Wiley & Sons, Ltd.     

Numerical accuracy of a Padé‐type non‐reflecting boundary condition for the finite element solution of acoustic scattering problems at high‐frequency

The present text deals with the numerical solution of two‐dimensional high‐frequency acoustic scattering problems using a new high‐order and asymptotic Padé‐type artificial boundary condition. The Padé‐type condition is easy‐to‐implement in a Galerkin least‐squares (iterative) finite element solver for arbitrarily convex‐shaped boundaries. The method accuracy is investigated for different model problems and for the scattering problem by a submarine‐shaped scatterer. As a result, relatively small computational domains, optimized according to the shape of the scatterer, can be considered while yielding accurate computations for high‐frequencies. Copyright © 2005 John Wiley & Sons, Ltd.     

Fully implicit Lagrange–Newton–Krylov–Schwarz algorithms for boundary control of unsteady incompressible flows

We develop a parallel fully implicit domain decomposition algorithm for solving optimization problems constrained by time‐dependent nonlinear partial differential equations. In particular, we study the boundary control of unsteady incompressible Navier–Stokes equations. After an implicit discretization in time, a fully coupled sparse nonlinear optimization problem needs to be solved at each time step. The class of full space Lagrange–Newton–Krylov–Schwarz algorithms is used to solve the sequence of optimization problems. Among optimization algorithms, the fully implicit full space approach is considered to be the easiest to formulate and the hardest to solve. We show that Lagrange–Newton–Krylov–Schwarz, with a one‐level restricted additive Schwarz preconditioner, is an efficient class of methods for solving these hard problems. To demonstrate the scalability and robustness of the algorithm, we consider several problems with a wide range of Reynolds numbers and time step sizes, and we present numerical results for large‐scale calculations involving several million unknowns obtained on machines with more than 1000 processors. Copyright © 2012 John Wiley & Sons, Ltd.     

Krylov precise time‐step integration method

An efficient precise time‐step integration (PTI) algorithm to solve large‐scale transient problems is presented in this paper. The Krylov subspace method and the Padé approximations are applied to modify the original PTI algorithm in order to improve the computational efficiency. Both the stability and accuracy characteristics of the resultant algorithms are investigated. The efficiency can be further improved by expanding the dimension to avoid the computation of the particular solutions. The present algorithm can also be extended to tackle nonlinear problems without difficulty. Two numerical examples are given to illustrate the highly accurate and efficient algorithm. Copyright © 2006 John Wiley & Sons, Ltd.     

Checkerboard‐free topology optimization using non‐conforming finite elements

The objective of the present study is to show that the numerical instability characterized by checkerboard patterns can be completely controlled when non‐conforming four‐node finite elements are employed. Since the convergence of the non‐conforming finite element is independent of the Lamé parameters, the stiffness of the non‐conforming element exhibits correct limiting behaviour, which is desirable in prohibiting the unwanted formation of checkerboards in topology optimization. We employ the homogenization method to show the checkerboard‐free property of the non‐conforming element in topology optimization problems and verify it with three typical optimization examples. Copyright © 2003 John Wiley & Sons, Ltd.     

A methodology for the generation of low‐cost higher‐order methods for linear dynamics

This work presents a methodology which generates efficient higher‐order methods for linear dynamics by improving the accuracy properties of Nørsett methods towards those of Padé methods. The methodology is based on a simple and low‐cost iterative procedure which is used to implement a set of higher‐order methods with controllable dissipation. A sequence of improved solutions is obtained which correspond to algorithms offering an effective compromise between the efficiency of Nørsett methods and the accuracy of Padé methods. Moreover, a direct control over high‐frequency dissipation is possible by means of an algorithmic parameter. Numerical tests are reported which confirm that this set of algorithms is really attractive for linear dynamic analysis. Copyright © 2003 John Wiley & Sons, Ltd.     

Topology optimization of dynamics problems with Padé approximants

An efficient procedure for topology optimization of dynamics problems is proposed. The method is based on frequency responses represented by Padé approximants and analytical sensitivity analysis derived using the adjoint method. This gives an accurate approximation of the frequency response over wide frequency ranges and a formulation that allows for design sensitivities to be computed at low computational cost also for a large number of design variables. Two examples that deal with optimization of forced vibrations are included. Copyright © 2007 John Wiley & Sons, Ltd.     

A human‐like numerical technique for design of engineering systems

Because of the necessity for considering various creative and engineering design criteria, optimal design of an engineering system results in a highly‐constrained multi‐objective optimization problem. Major numerical approaches to such optimal design are to force the problem into a single objective function by introducing unjustifiable additional parameters and solve it using a single‐objective optimization method. Due to its difference from human design in process, the resulting design often becomes completely different from that by a human designer. This paper presents a novel numerical design approach, which resembles the human design process. Similar to the human design process, the approach consists of two steps: (1) search for the solution space of the highly‐constrained multi‐objective optimization problem and (2) derivation of a final design solution from the solution space. Multi‐objective gradient‐based method with Lagrangian multipliers (MOGM‐LM) and centre‐of‐gravity method (CoGM) are further proposed as numerical methods for each step. The proposed approach was first applied to problems with test functions where the exact solutions are known, and results demonstrate that the proposed approach can find robust solutions, which cannot be found by conventional numerical design approaches. The approach was then applied to two practical design problems. Successful design in both the examples concludes that the proposed approach can be used for various design problems that involve both the creative and engineering design criteria. Copyright © 2005 John Wiley & Sons, Ltd.     

A boundary condition in Padé series for frequency‐domain solution of wave propagation in unbounded domains

A boundary condition satisfying the radiation condition at infinity is frequently required in the numerical simulation of wave propagation in an unbounded domain. In a frequency domain analysis using finite elements, this boundary condition can be represented by the dynamic stiffness matrix of the unbounded domain defined on its boundary. A method for determining a Padé series of the dynamic stiffness matrix is proposed in this paper. This method starts from the scaled boundary finite‐element equation, which is a system of ordinary differential equations obtained by discretizing the boundary only. The coefficients of the Padé series are obtained directly from the ordinary differential equations, which are not actually solved for the dynamic stiffness matrix. The high rate of convergence of the Padé series with increasing order is demonstrated numerically. This technique is applicable to scalar waves and elastic vector waves propagating in anisotropic unbounded domains of irregular geometry. It can be combined seamlessly with standard finite elements. Copyright © 2006 John Wiley & Sons, Ltd.     

Weighting parameters for time‐step integration algorithms with predetermined coefficients

In this paper, the effect of using the predetermined coefficients in constructing time‐step integration algorithms is investigated. Both first‐ and second‐order equations are considered. The approximate solution is assumed to be in a form of polynomial in the time domain. It can be related to the truncated Taylor's series expansion of the exact solution. Therefore, some of the coefficients can be predetermined from the known initial conditions. If there are m predetermined coefficients and r unknown coefficients in the approximate solution, the unknowns can be solved by the weighted residual method. The weighting parameter method is used to investigate the resultant algorithm characteristics. It is shown that the formulation is consistent with a minimum order of accuracy m+r. The maximum order of accuracy achievable is m+2r. Unconditionally stable algorithms exist if m⩽r for first‐order equations and m+1⩽r for second‐order equations. Hence, the Dahlquist's theorem is generalized. Algorithms equivalent to the Padé approximations and unconditionally stable algorithms equivalent to the generalized Padé approximations are constructed. The corresponding weighting parameters and weighting functions for the Padé and generalized Padé approximations are given explicitly. Copyright © 2000 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.     

Weighting parameters for unconditionally stable higher‐order accurate time step integration algorithms. Part 1—first‐order equations

In this paper, unconditionally stable higher‐order accurate time step integration algorithms suitable for linear first‐order differential equations based on the weighted residual method are presented. Instead of specifying the weighting functions, the weighting parameters are used to control the algorithm characteristics. If the numerical solution is approximated by a polynomial of degree n, the approximation is at least nth‐order accurate. By choosing the weighting parameters carefully, the order of accuracy can be improved. The generalized Padé approximations with polynomials of degree n as the numerator and denominator are considered. The weighting parameters are chosen to reproduce the generalized Padé approximations. Once the weighting parameters are known, any set of linearly independent basic functions can be used to construct the corresponding weighting functions. The stabilizing weighting factions for the weighted residual method are then found explicitly. The accuracy of the particular solution due to excitation is also considered. It is shown that additional weighting parameters may be required to maintain the overall accuracy. The corresponding equations are listed and the additional weighting parameters are solved explicitly. However, it is found that some weighting functions could satisfy the listed equations automatically. Copyright © 1999 John Wiley & Sons, Ltd.     

A phase field model for stress‐based evolution of load‐bearing structures

We present a continuum‐type optimality algorithm for the evolution of load‐bearing solid structures with linear elastic material. The objective of our model is to generate structures with help of a sensitivity function accounting for equivalent stress. Similar to Evolutionary Structural Optimization, a threshold of equivalent stress is evaluated. However, we do not consider a material rejection ratio. The evolution process is governed by an Allen‐Cahn equation in the context of phase field modeling. The steady state of phase transition is the final geometry of the structure. The model accounts for the desired filling level, geometry of the design space, static loadings, and boundary conditions. Our variational approach drops conservation of mass and couples density as well as stiffness of the continuum by a sophisticated function to the phase field variable. Continuous regions of voids and material evolve from the initial state of homogeneously distributed material. The complexity of evolving structures depends on two numerical parameters, which we discuss in several examples.     

Fast frequency sweep computations using a multi‐point Padé‐based reconstruction method and an efficient iterative solver

Problems of the form Z (σ) u (σ)= f (σ), where Z is a given matrix, f is a given vector, and σ is a circular frequency or circular frequency‐related parameter arise in many applications including computational structural and fluid dynamics, and computational acoustics and electromagnetics. The straightforward solution of such problems for fine increments of σ is computationally prohibitive, particularly when Z is a large‐scale matrix. This paper discusses an alternative solution approach based on the efficient computation of u and its successive derivatives with respect to σ at a few sample values of this parameter, and the reconstruction of the solution u (σ) in the frequency band of interest using multi‐point Padé approximants. This computational methodology is illustrated with applications from structural dynamics and underwater acoustic scattering. In each case, it is shown to reduce the CPU time required by the straightforward approach to frequency sweep computations by two orders of magnitude. Copyright © 2006 John Wiley & Sons, Ltd.     

Padé extensions to ray series methods and analysis of transients in inhomogeneous viscoelastic media of hereditary type

In this paper we demonstrate that wavefront expansions for the analysis of transient phenomena are far from adequate when numerical information back of the wavefront is required. However, by employing Padé approximants together with ray series methods, we can obtain directly and greatly extend the range of validity of these expansions. The procedure, which is straightforward and requires very little computing time, is here applied to a non-trivial problem involving impact-generated shear transients in inhomogenoues viscoelastic media whose stress-strain laws are given in integral form. For a special combination of the material parameters an exact solution is recovered and used to check the validity of our approximate Padé-extended wavefront solution. We also compare our results from the extended wavefront solution with numerical solutions obtained using Bellman's approximate inversion scheme for Laplace transforms. A further advantage of our approach is that, unlike transform techniques, it does not depend upon being able to find tabulated special function equations for the transformed dependent variables. All numerical results are presented graphically for ease of comparison.     

Design space optimization using a numerical design continuation method

A generalized optimization problem in which design space is also a design to be found is defined and a numerical implementation method is proposed. In conventional optimization, only a portion of structural parameters is designated as design variables while the remaining set of other parameters related to the design space are often taken for granted. A design space is specified by the number of design variables, and the layout or configuration. To solve this type of design space problems, a simple initial design space is selected and gradually improved while the usual design variables are being optimized. To make the design space evolve into a better one, one may increase the number of design variables, but, in this transition, there are discontinuities in the objective and constraint functions. Accordingly, the sensitivity analysis methods based on continuity will not apply to this discontinuous stage. To overcome the difficulties, a numerical continuation scheme is proposed based on a new concept of a pivot phase design space. Two new categories of structural optimization problems are formulated and concrete examples shown. Copyright © 2001 John Wiley & Sons, Ltd.     

Convergence speed in multi‐objective metaheuristics: Efficiency criteria and empirical study

Solving optimization problems using a reduced number of objective function evaluations is an open issue in the design of multi‐objective optimization metaheuristics. The usual approach to analyze the behavior of such techniques is to choose a benchmark of known problems, to perform a predetermined number of function evaluations, and then, apply a set of performance indicators in order to assess the quality of the solutions obtained. However, this sort of methodology does not provide any insights of the efficiency of each algorithm. Here, efficiency is defined as the effort required by a multi‐objective metaheuristic to obtain a set of non‐dominated solutions that is satisfactory to the user, according to some pre‐defined criterion. Indeed, the type of solutions of interest to the user may vary depending on the specific characteristics of the problem being solved. In this paper, the convergence speed of seven state‐of‐the‐art multi‐objective metaheuristics is analyzed, according to three pre‐defined efficiency criteria. Our empirical study shows that SMPSO (based on a particle swarm optimizer) is found to be the best overall algorithm on the test problems adopted when considering the three efficiency criteria. Copyright © 2010 John Wiley & Sons, Ltd.