Bicriterial Design of Digital Filters

A problem of technical interest is the solution of approximation problems which make a tradeoff between the L ₂ norm and the L norm error criteria. This problem is investigated in the framework of filter design with respect to two conflicting optimality goals. The particular interest in L ₂-L norm compromise filters has been raised by a paper of Adams (IEEE Trans. on Circuits and Systems vol. 39, pp. 376–388, 1991), who suggested to compute such FIR filters by solution of certain constrained L ₂ approximation problems which require a proper choice of weights. It is shown in this paper that bicriterial filter design problems can be approached by classical methods from multicriteria optimization and that especially reference point approximation with the ideal point as reference point is a suitable tool to deal with Adams' problem. Solutions from this latter approach do especially not depend on the choice of weights and yield the best possible compromise filters with respect to a prescribed measure. The resulting optimization problems can be solved with (semi-infinite) programming methods having proven convergence under standard assumptions. Examples of L ₂-L norm compromise designs of a linear-phase FIR and an IIR filter are presented.     

Robust H 8 output feedback control for general nonlinear systems with structured uncertainty

Abstract

In this paper, the robust H ⁸ output feedback control problem for general nonlinear systems with L ₂‐norm‐bounded structured uncertainties is considered. Sufficient conditions for the solvability of robust performance synthesis problems are represented in terms of two Hamilton‐Jacobi inequalities with n independent variables. Based on these conditions, a state space characterization of a robust H ⁸ output feedback controller solving the considered problem is proposed. An example is provided for illustration.     

Current Density Reconstructions for Lorentz Force Evaluation

The detection and reconstruction of fatigue fractures is of great interest in quality assurance. In the framework of nondestructive testing, Lorentz force evaluation (LFE) is an evaluation technique to estimate flaws in electrically conductive materials based on measured Lorentz forces. In the forward solution for LFE, a defect can be interpreted as a distributed current source. This has motivated the authors to propose current density reconstructions (CDRs) calculated with minimum norm estimates to estimate defect geometries. The L₁ and L₂ norms tend to produce a solution which is either very focused or very smeared. To balance these constraints, the general L_p norm with 1 ≤ p ≤ 2 was used and the inverse solutions compared. This approach was applied to measured data obtained from a laminated composite and simulated data from a monolithic material. The results show that the L_1.5 norm provides the most accurate inverse solutions. The location and extent of the defect are determined with an error of 15 % relative to the size of the defect. The depth estimation has a deviation of 50 %. It can be concluded that CDRs are a powerful method to reconstruct and characterize defects in LFE.     

Solution of low péclet number heat transfer with viscous dissipation in the semi‐infinite axial region of a tube via functional analytic methods

Abstract

A solution of the extended Graetz problem with prescribed wall heat flux and viscous dissipation in a semi‐infinite axial region of a tube is obtained by functional analytic methods. The energy equation is split into a set of partial differential equations to obtain a self‐adjoint formulism. Then, an algebraic characteristic equation of the eigenvalue problem for an arbitrary velocity profile is obtained by an approximation method in L ₂[0, 1]. In addition, a backward recursive formula for calculating the expansion coefficients of the solution is developed.     

A residual a posteriori error estimate for partition of unity finite elements for three-dimensional transient heat diffusion problems using multiple global enrichment functions

In this article, a study of residual based a posteriori error estimation is presented for the partition of unity finite element method (PUFEM) for three-dimensional (3D) transient heat diffusion problems. The proposed error estimate is independent of the heuristically selected enrichment functions and provides a useful and reliable upper bound for the discretization errors of the PUFEM solutions. Numerical results show that the presented error estimate efficiently captures the effect of h-refinement and q-refinement on the performance of PUFEM solutions. It also efficiently reflects the effect of ill-conditioning of the stiffness matrix that is typically experienced in the partition of unity based finite element methods. For a problem with a known exact solution, the error estimate is shown to capture the same solution trends as obtained by the classical L₂ norm error. For problems with no known analytical solutions, the proposed estimate is shown to be used as a reliable and efficient tool to predict the numerical errors in the PUFEM solutions of 3D transient heat diffusion problems.     

A new achievement scalarizing function based on parameterization in multiobjective optimization

This paper addresses a general multiobjective optimization problem. One of the most widely used methods of dealing with multiple conflicting objectives consists of constructing and optimizing a so-called achievement scalarizing function (ASF) which has an ability to produce any Pareto optimal or weakly/properly Pareto optimal solution. The ASF minimizes the distance from the reference point to the feasible region, if the reference point is unattainable, or maximizes the distance otherwise. The distance is defined by means of some specific kind of a metric introduced in the objective space. The reference point is usually specified by a decision maker and contains her/his aspirations about desirable objective values. The classical approach to constructing an ASF is based on using the Chebyshev metric L _∞. Another possibility is to use an additive ASF based on a modified linear metric L ₁. In this paper, we propose a parameterized version of an ASF. We introduce an integer parameter in order to control the degree of metric flexibility varying from L ₁ to L _∞. We prove that the parameterized ASF supports all the Pareto optimal solutions. Moreover, we specify conditions under which the Pareto optimality of each solution is guaranteed. An illustrative example for the case of three objectives and comparative analysis of parameterized ASFs with different values of the parameter are given. We show that the parameterized ASF provides the decision maker with flexible and advanced tools to detect Pareto optimal points, especially those whose detection with other ASFs is not straightforward since it may require changing essentially the reference point or weighting coefficients as well as some other extra computational efforts.     

Computation of Single Eigenfrequencies and Eigenfunctions of Plate and Shell Structures Using an H-adaptive FE-method

The accuracy of the computation of eigenfrequencies and eigenfunctions with FE-methods can be substantially improved with efficient adaptive procedures. For such an adaptive analysis of plate and shell structures a simple a-posteriori error estimator or indicator for the error in the energy norm and L ₂-norm of the eigenfunction u _h for shell and plate structures is proposed.Both indicators hite shall represent the correct convergence of the estimated error. The estimator for the error in the energy norm is used to enlarge adaptively the dimension of the finite element subspace. On the basis of numerical examples the efficiency and the quality of the improved solution is discussed. In order to validate the quantity of the estimated error Aitken’s extrapolation technique is applied.     

Generalised critical free-surface flows

Nonlinear waves in a forced channel flow are considered. The forcing is due to a bottom obstruction. The study is restricted to steady flows. A weakly nonlinear analysis shows that for a given obstruction, there are two important values of the Froude number, which is the ratio of the upstream uniform velocity to the critical speed of shallow water waves, F _C>1 and F _L<1 such that: (i) when F<F _L, there is a unique downstream cnoidal wave matched with the upstream (subcritical) uniform flow; (ii) when F=F _L, the period of the cnoidal wave extends to infinity and the solution becomes a hydraulic fall (conjugate flow solution) – the flow is subcritical upstream and supercritical downstream; (iii) when F>F _C, there are two symmetric solitary waves sustained over the site of forcing, and at F=F _C the two solitary waves merge into one; (iv) when F>F _C, there is also a one-parameter family of solutions matching the upstream (supercritical) uniform flow with a cnoidal wave downstream; (v) for a particular value of F>F _C, the downstream wave can be eliminated and the solution becomes a reversed hydraulic fall (it is the same as solution (ii), except that the flow is reversed!). Flows of type (iv), including the hydraulic fall (v) as a special case, are computed here using the full Euler equations. The problem is solved numerically by a boundary-integral-equation method due to Forbes and Schwartz. It is confirmed that there is a three-parameter family of solutions with a train of waves downstream. The three parameters can be chosen as the Froude number, the obstruction size and the wavelength of the downstream waves. This three-parameter family differs from the classical two-parameter family of subcritical flows (i) but includes as a particular case the hydraulic falls (ii) or equivalently (v) computed by Forbes.     

On liouvillian solutions of linear differential equations

This paper deals with the problem of finding liouvillian solutions of a homogeneous linear differential equationL(y)=0 of ordern with coefficients in a differential fieldk. For second order linear differential equations with coefficients ink _o(x), wherek _o is a finite algebraic extension ofQ, such an algorithm has been given by J. Kovacic and implemented. A general decision procedure for finding liouvillian solutions of a differential equation of ordern has been given by M.F. Singer, but the resulting algorithm, although constructive, is not in implementable form even for second order equations. Both algorithms use the fact that, ifL(y)=0 has a liouvillian solution, then,L(y)=0 has a solutionz such thatu=z/z is algebraic overk. Using the action of the differential galois group onu and the theory of projective representation we get sharp bounds (n) for the algebraic degree ofu for differential equations of arbitrary ordern. For second order differential equations we get the bound (2)=12 used in the algorithm of J. Kovacic and for third order differential equation we improve the bound given by M.F. Singer from 360 to (3)36. We also show that not all values less than or equal to (n) are possible values for the algebraic degree ofu. For second order differential equations we rediscover the values 2, 4, 6, and 12 used in the Kovacic Algorithm and for third order differential equations we get the possibilities 3,4, 6, 7, 9, 12, 21, and 36. We prove that if the differential Galois group ofL(y)=0 is a primitive unimodular linear group, then all liouvillian solutions are algebraic. From this it follows that, if a third order differential equationL(y)=0 is not of Fuchsian type, then the logarithmic derivative of some liouvillian solution ofL(y)=0 is algebraic of degree 3. We also derive an upper bound for the minimal numberN(n) of possible degreesm of the minimal polynomial of an algebraic solution of the riccati equation associated withL(y)=0.Supported by Deutsche Forschungsgemeinschaft while visiting North Carolina State University     

A multiple-scales approach to crack-front waves

Perturbation of a steadily propagating crack with a straight edge is solved using the method of matched asymptotic expansions (MAE). This provides a simplified analysis in which the inner and outer solutions are governed by distinct mechanics. The inner solution contains the explicit perturbation and is governed by a quasi-static equation. The outer solution determines the radiation of energy away from the tip, and requires solving dynamic equations in the unperturbed configuration. The outer and inner expansions are matched via the small parameter ϵ = L/l defined by the disparate length scales: the crack perturbation length L and the outer length scale l associated with the loading. The method is illustrated for a scalar crack model and then applied to the elastodynamic mode I problem. The crack-front wave-dispersion relation is found by requiring that the energy release rate is unaltered under perturbation and dispersive properties of the crack-front wave speed are described for the first time. The example problems considered demonstrate the potential of MAE for moving-boundary-value problems with multiple scales.     

Inverse design of directional solidification processes in the presence of a strong external magnetic field

A computational method for the design of directional alloy solidification processes is addressed such that a desired growth velocity ν_f under stable growth conditions is achieved. An externally imposed magnetic field is introduced to facilitate the design process and to reduce macrosegregation by the damping of melt flow. The design problem is posed as a functional optimization problem. The unknowns of the design problem are the thermal boundary conditions. The cost functional is taken as the square of the L₂ norm of an expression representing the deviation of the freezing interface thermal conditions from the conditions corresponding to local thermodynamic equilibrium. The adjoint method for the inverse design of continuum processes is adopted in this work. A continuum adjoint system is derived to calculate the adjoint temperature, concentration, velocity and electric potential fields such that the gradient of the L₂ cost functional can be expressed analytically. The cost functional minimization process is realized by the conjugate gradient method via the FE solutions of the continuum direct, sensitivity and adjoint problems. The developed formulation is demonstrated with an example of designing the boundary thermal fluxes for the directional growth of a germanium melt with dopant impurities in the presence of an externally applied magnetic field. The design is shown to achieve a stable interface growth at a prescribed desired growth rate. Copyright © 2001 John Wiley & Sons, Ltd.     

A Padé‐based factorization‐free algorithm for identifying the eigenvalues missed by a generalized symmetric eigensolver

When computing the solution of a generalized symmetric eigenvalue problem of the form Ku =λ Mu , the Sturm sequence check, also known as the inertia check, is the most popular method for reporting the number of missed eigenvalues within a range [σ_L,σ_R]. This method requires the factorization of the matrices K ?σ_L M and K ?σ_R M . When the size of the problem is reasonable and the matrices K and M are assembled, these factorizations are possible. When the eigensolver is equipped with an iterative solver, which is nowadays the preferred choice for large‐scale problems, the factorization of K ?σ M is not desired or feasible and therefore the inertia check cannot be performed. To this effect, the purpose of this paper is to present a factorization‐free algorithm for detecting and identifying the eigenvalues that were missed by an eigensolver equipped with an iterative linear equation solver within an interval of interest [σ_L,σ_R]. This algorithm constructs a scalar, rational, transfer function whose poles are exactly the eigenvalues of the symmetric pencil ( K , M ), approximates it by a Padé expansion, and computes the poles of this approximation to detect and identify the missed eigenvalues. The proposed algorithm is illustrated with an academic numerical example. Its potential for real engineering applications is also demonstrated with a large‐scale structural vibrations problem. Copyright © 2009 John Wiley & Sons, Ltd.     

Online dynamic cardiac imaging based on the elastic-net model

Purpose: The purpose of this work was to develop an online dynamic cardiac MRI model to reconstruct image frames from partial acquisition of the Cartesian k-space data, which utilizes structural knowledge of consecutive image frames. Materials and methods: Using an elastic-net model, the proposed algorithm reconstructs dynamic images using both L₁ and L₂ norm operations. The L₁ norm enforces the sparsity of the frame difference, while the L₂ norm with motion-adaptive weights catches the internal structure of frame differences. Unlike other online methods such as the Kalman filter (KF) technique, the new model requires no assumption of Gaussian noise, and can faithfully reconstruct the dynamic images within a compressive sensing framework. Results: The proposed method was evaluated using simulated dynamic phantoms with 40 frames of images (128?×?128) and a cardiac MRI cine of 25 frames (256?×?256). Both results showed that the new model offered a better performance than the online KF method in depicting simulated phantom and cardiac dynamics. Conclusion: It is concluded that the proposed imaging model can be used to capture a large variety of objects in motion from highly under-sampled k-space data, and being particularly useful for improving temporal resolution of cardiac MRI.     

Optimal modal reduction of dynamic subsystems: Extensions and improvements

The problem of optimal reduction of a linear dynamic subsystem is revisited. The subsystem may represent, for example, a complex but minor part of a large elastic structure. The goal is to drastically reduce the number of degrees of freedom of a subsystem attached through an interface to a main system in such a way as to affect the dynamic behavior of the main system in the least possible way. In a recent publication, the Optimal Modal Reduction (OMR) algorithm was developed to this end. This algorithm seeks a reduction of the subsystem that will have minimal effect, in the L₂ norm, on the Dirichlet‐to‐Neumann (DtN) map on the interface. Here this algorithm is extended and improved in a number of ways. First, a family of alternative formulations are derived for the DtN map, which lead to alternative OMR algorithms; one of them, called OMR _j=2, is shown to yield better results than the original formulation. Second, the OMR algorithm, which was originally developed for undamped subsystems, is extended to subsystems undergoing Rayleigh damping. In addition, an enhanced derivation of the original OMR algorithm is presented, and the good pointwise performance of OMR is explained by relating to minimization with respect to the H¹ norm. The extensions mentioned above are discussed theoretically as well as demonstrated via numerical examples. Experiments and discussion include comparison of the OMR algorithm to simple coarsening of the subsystem discretization. In all cases, central finite difference discretization in space and explicit time‐stepping are employed to solve the scalar wave equation. Copyright © 2010 John Wiley & Sons, Ltd.     

Finite horizon H∞ filter and its 2N algorithm

A 2^N algorithm will double a time (or space) step in each evaluation for initial value problem. The 2^N algorithm for the integration of filtering differential equation of the finite horizon H_∞ filter is presented in this paper. Since it is a boundary value problem within a time range, a new 2^N algorithm is introduced by merging two intervals each time so that the time interval is doubled in each evaluation. If one divides the original time range into one million intervals, 20 evaluations will complete the whole process. Owing to the extremely small initial time interval, the first few terms of the Taylor expansion of the interval matrices are sufficient for very accurate results. Since the filter gain matrices are the solution of the Riccati differential equation and the existence of the solution depends on the induced norm γ, the computation of critical value γ is reviewed first. Then, according to the result and the prespecified performance index, the suitable parameter γ^?2 can be selected and the precise numerical solution of the Riccati differential equation and the filtering differential equation can be obtained by using the 2^N algorithm, although the filtering equation is time varying. The 2^N algorithm for interval merging is given explicitly. Copyright © 2001 John Wiley & Sons, Ltd.     

The boundary-element method for the determination of a heat source dependent on one variable

This paper investigates the inverse problem of determining a heat source in the parabolic heat equation using the usual conditions of the direct problem and a supplementary condition, called an overdetermination. In this problem, if the heat source is taken to be space-dependent only, then the overdetermination is the temperature measurement at a given single instant, whilst if the heat source is time-dependent only, then the overdetermination is the transient temperature measurement recorded by a single thermocouple installed in the interior of the heat conductor. These measurements ensure that the inverse problem has a unique solution, but this solution is unstable, hence the problem is ill-posed. This instability is overcome using the Tikhonov regularization method with the discrepancy principle or the L-curve criterion for the choice of the regularization parameter. The boundary-element method (BEM) is developed for solving numerically the inverse problem and numerical results for some benchmark test examples are obtained and discussed     

On error estimates and adaptivity in elastoplastic solids: Applications to the numerical simulation of strain localization in classical and Cosserat continua

The a posteriori error estimates based on the post-processing approach are introduced for elastoplastic solids. The standard energy norm error estimate established for linear elliptic problems is generalized here to account for the presence of internal variables through the norm associated with the complementary free energy. This is known to represent a natural metric for the class of elastoplastic problems of evolution. In addition, the intrinsic dissipation functional is utilized as a basis for a complementary a posteriori error estimates. A posteriori error estimates and adaptive refinement techniques are applied to the finite element analysis of a strain localization problem. As a model problem, the constitutive equations describing a generalization of standard J₂-elastoplasticity within the Cosserat continuum are used to overcome serious limitations exhibited by classical continuum models in the post-instability region. The proposed a posteriori error estimates are appropriately modified to account for the Cosserat continuum model and linked with adaptive techniques in order to simulate strain localization problems. Superior behaviour of the Cosserat continuum model in comparison to the classical continuum model is demonstrated through the finite element simulation of the localization in a plane strain tensile test for an elastopiastic softening material, resulting in convergent solutions with an h-refinement and almost uniform error distribution in all considered error norms.     

The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique

This is the first of two papers concerning superconvergent recovery techniques and a posteriori error estimation. In this paper, a general recovery technique is developed for determining the derivatives (stresses) of the finite element solutions at nodes. The implementation of the recovery technique is simple and cost effective. The technique has been tested for a group of widely used linear, quadratic and cubic elements for both one and two dimensional problems. Numerical experiments demonstrate that the recovered nodal values of the derivatives with linear and cubic elements are superconvergent. One order higher accuracy is achieved by the procedure with linear and cubic elements but two order higher accuracy is achieved for the derivatives with quadratic elements. In particular, an O(h⁴) convergence of the nodal values of the derivatives for a quadratic triangular element is reported for the first time. The performance of the proposed technique is compared with the widely used smoothing procedure of global L₂ projection and other methods. It is found that the derivatives recovered at interelement nodes, by using L₂ projection, are also superconvergent for linear elements but not for quadratic elements. Numerical experiments on the convergence of the recovered solutions in the energy norm are also presented. Higher rates of convergence are again observed. The results presented in this part of the paper indicate clearly that a new, powerful and economical process is now available which should supersede the currently used post-processing procedures applied in most codes.     

Transonic flow past thin wings

This paper is devoted to a discussion of steady inviscid transonic flow past thin wings, with subsonic free-stream Mach number M_∞ < 1, by the integral equation method. The integral equation formulation is developed for a thin unsymmetric wing at small incidence. A simple approximate analytical solution is presented for shock-free supercritical flow past a thin symmetric wing at zero incidence. The direct iteration scheme of Niyogi and Chakraborty is then extended to the three-dimensional zero incidence case, which may be used to obtain more accurate solutions for shockfree flows as well as for flows with shocks. The question of the existence and the uniqueness of a solution has been studied by means of the Banach contraction mapping principle in the spaceL ₂ (E₃), which establishes the condition of convergence of the direct iteration scheme. Simultaneously it provides us with an error estimate for the solution.