Equilibrated patch recovery error estimates: simple and accurate upper bounds of the error

This paper introduces a new recovery‐type error estimator ensuring local equilibrium and yielding a guaranteed upper bound of the error. The upper bound property requires the recovered solution to be both statically equilibrated and continuous. The equilibrium is obtained locally (patch‐by‐patch) and the continuity is enforced by a postprocessing based on the partition of the unity concept. This postprocess is expected to preserve the features of the locally equilibrated stress field. Nevertheless, the postprocess phase modifies the equilibrium, which is no longer exactly fulfilled. A new methodology is introduced that yields upper bound estimates by taking into account this lack of equilibrium. This requires computing the ??₂ norm of the error or relating it with the energy norm. The guaranteed upper bounds are obtained by using a pessimistic bound of the error ??₂ norm, derived from an eigenvalue problem. Nevertheless, these bounds are not sharp. An additional strategy based on a more accurate assessment of the error ??₂ norm is introduced, providing sharp estimates, which are practical upper bounds as it is demonstrated in the numerical tests. Copyright © 2006 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.     

Computation of upper and lower bounds to the frequencies of elastic systems by the method of Lehmann and Maehly

A method, called the Lehmann-Maehly method, for determining upper and lower bounds for eigenvalues, which was introduced by Bazley and Fox⁴ in 1964, is applied to the determination of bounds to the frequencies of elastic systems. In reviewing the fundamental theory involved, the present paper emphasizes the physical significance of the adjoint operators T and T* employed. An error in the paper by Bazley and Fox is noted and it is pointed out that with the corrected theory it is possible to use the Lehmann-Maehly method as a procedure for computing converging sequences of upper and lower bounds merely by varying a constant, the shifting constant. Thus, sequences of bounds are shown to be easily obtainable without prior knowledge of rough bounds, and the procedure for obtaining the bounds is not significantly more difficult to apply than the familiar Rayleigh-Ritz method for upper bounds. Since displacements and stresses are independently varied, the displacement functions used in the approximation procedure are only required to satisfy prescribed displacement boundary conditions while the stress functions only need satisfy the natural (stress) boundary conditions. Operators are derived for a non- uniform beam based on Timoshenko beam theory. Tables of bounds computed by the Lehmann-Maehly method and also the Rayleigh-Ritz method are given as an illustrative example.     

A posteriori finite element bounds to linear functional outputs of the three‐dimensional Navier–Stokes equations

An implicit a posteriori finite element error estimation method is presented to inexpensively calculate lower and upper bounds for a linear functional output of the numerical solutions to the three‐dimensional Navier–Stokes (N–S) equations. The novelty of this research is to utilize an augmented Lagrangian based on a coarse mesh linearization of the N–S equations and the finite element tearing and interconnecting (FETI) procedure. The latter approach extends the a posteriori bound method to the three‐dimensional Crouzeix–Raviart space for N–S problems. The computational advantage of the bound procedure is that a single coupled non‐symmetric large problem can be decomposed into several uncoupled symmetric small problems. A simple model problem, which is selected here to illustrate the procedure, is to find the lower and upper bounds of the average velocity of a pressure driven, incompressible, steady Newtonian fluid flow moving at low Reynolds numbers through an endless square channel which has an array of rectangular obstacles. Numerical results show that the bounds for this output are rigorous, i.e. always in the asymptotic certainty regime, that they are sharp and that the required computational resources decrease significantly. Parallel implementation on a Beowulf cluster is also reported. Copyright © 2004 John Wiley & Sons, Ltd.     

Some Remarks on a General Class of Markov Processes with Discrete Time Parameter and Dependent Increments

In this paper we consider a general class of Markov processes with discrete time parameter {X _n : n = 0, 1, 2, …}, where the transitions from one state to another are determined by:

The functions h _j (·) are strictly increasing, and in general the Markov processes have dependent increments. In section 1 the model is described more precisely. Considering the sequence of distribution functions F _n(x) = Pr[X _n ≤ x] (n = 0, 1, 2, ….) a numerical method is given by which upper and lower bounds can be determined for F _n(x) (n ≥ n _o). Numerical results are given for some special cases, and the determination of an alarm level in a flame failure control system is described as an application. The results show that the upper and lower bounds can lie very close together.     

不可约Z-矩阵最小特征值的数值算法

首先给出了不可约非负矩阵最大特征值的上下界。然后利用相似变换构造了一列相似矩阵,从而得到不可约非负矩阵最大特征值的逐步压缩的一列上下界,其极限为所要求的最大特征值。最后利用Z-矩阵与非负矩阵的关系,给出了计算不可约Z-矩阵最小特征值的一个新算法。理论上给出了收敛性证明。该算法迭代过程简单,不用计算逆矩阵,从而计算量小,占用内存少。数值实验的结果表明该算法具有可行性和有效性。     

Application of the Krein's method for determination of natural frequencies of periodically supported beam based on simplified Bresse-Timoshenko equations

Summary A new method is proposed for the stress analysis of an elastic space weakened by several arbitrarily located coplanar circular cracks under the action of an arbitrary normal pressure. The method is based on a new type of integral equations which has definite advantages over the existing methods: equations are non-singular, the iteration procedure is rapidly convergent even for very close interactions; there is no need to solve the integral equations if one is interested only in obtaining the upper and the lower bounds for the quantities of interest. In the case of the cracks far apart, these bounds are so close that provide, in fact, a sufficiently accurate solution to the problem. The method allows us also to obtain a practically exact numerical solution to the problem of very close interactions. Several illustrative examples are considered.     

Optimizing Two-Level Supersaturated Designs Using Swarm Intelligence Techniques

Supersaturated designs (SSDs) are often used to reduce the number of experimental runs in screening experiments with a large number of factors. As more factors are used in the study, the search for an optimal SSD becomes increasingly challenging because of the large number of feasible selection of factor level settings. This article tackles this discrete optimization problem via an algorithm based on swarm intelligence. Using the commonly used E(s²) criterion as an illustrative example, we propose an algorithm to find E(s²)-optimal SSDs by showing that they attain the theoretical lower bounds found in previous literature. We show that our algorithm consistently produces SSDs that are at least as efficient as those from the traditional CP exchange method in terms of computational effort, frequency of finding the E(s²)-optimal SSD, and also has good potential for finding D₃-, D₄-, and D₅-optimal SSDs. Supplementary materials for this article are available online.     

Lower confidence bounds for C PU and C PL based on multiple samples with application to production yield assurance

For stably normal processes with one-sided specification limits, capability indices C _PU and C _PL have been used to provide numerical measures on production yield assurance. Statistical properties of the estimators of C _PU and C _PL have been investigated extensively for cases with a single sample. It is shown that for multiple samples, the uniformly minimum-variance unbiased estimators of C _PU and C _PL are consistent and asymptotically efficient. Based on the uniformly minimum-variance unbiased estimators, an algorithm is developed with an efficient program using a direct search method to compute the lower confidence bounds for C _PU and C _PL. The lower confidence bounds convey critical information to the minimum capability of a process, providing a necessary yield assurance of production. The lower confidence bounds are tabulated for some commonly used capability requirement so that engineers/practitioners can use them for their in-plant applications. An example of a high-speed buffer amplifier is presented to illustrate the practicality of the approach to data collected from the factories for production yield assurance.     

Bilateral bounds for the shear and torsion factors: comments on elementary derivations

This paper proposes a new simple derivation of bilateral bounds for the strain energy–based shear and torsion factors, χ _i , of an elastic beam together with some comments about the coherence of the current formulations. A rearrangement of the definition as a mean over the cross-section and an original decomposition of the shear stress in two parts—τ _eqv that is equivalent to the external force and a residual Δτ—allow (i) to interpret (χ − 1) as the mean quadratic deviation of the shear field with respect to the distribution τ _eqv and (ii) to easily evaluate an upper bound, using minimal information about the stress field. In this formulation, the lower bound becomes trivial. Several numerical examples illustrate the accuracy and suitability of the results obtained.     

An efficient method for solving the eigenvalue problem for matrices having a skew-symmetric (or skew-Hermitian) component of low rank

In the numerical modelling of mechanical systems, eigenvalue problems occur in connection with the evaluation of resonance frequencies, buckling modes and other more esoteric calculations. The matrices whose eigenvalues are sought sometimes have a skew-symmetric component and the presence of this component adds significantly to the computational effort required. In many cases where there is a skew-symmetric component, this component has a much lower rank than the symmetric component which generally has rank equal to its dimension. Examples of such cases abound in the area of rotor dynamics where the stiffness and damping matrices associated with journal bearings have significant skew-symmetric components. The solution of the eigenvalue problem for an unsymmetric matrix takes more than twice the number of operations required for the solution of the eigenvalue problem for a symmetric matrix of the same dimension. This paper puts forward a new method for the solution of the eigenvalue problem for matrices having a skew-symmetric component of low rank and shows that it is faster than established methods of comparable accuracy for the general unsymmetric NxN matrix if the rank of the skew-symmetric component is less than N/7.3.     

Binomial Group-Testing With an Unknown Proportion of Defectives

The binomial group-testing problem is extended to the case in which the common probability p of a unit being defective is unknown. A Bayes “non-mixing” procedure R ⁽¹⁾ is derived and compared with other procedures, in particular with the corresponding procedure R ₁ that requires the knowledge of p and with another procedure based on continually revising the maximum likelihood estimate of p; the latter is called an empirical Bayes solution. Finally, a Bayes procedure derived in the Appendix allows “mixing” and this is conjectured to be the unrestricted Bayes solution; the improvements due to “mixing” are shown to be small in Table III. Several applications of the general problem of group-testing are discussed in the introduction and this virtually constitutes a general review of the subject. After the procedure R ⁽¹⁾ is defined a detailed illustration is given in section 2.1 showing how to carry out this procedure with the use of Tables I and II. Lower bounds for the Bayes risk associated with any group testing procedure for unknown p are given in Table III; one of these bounds, based on information theory, is derived in section 4.     

The Edge Effect of a Microstrip Resonator on the Field Dependence of the Surface Resistance of a High-Tc Superconducting Thin Film

A new behavior of the field dependence of microwave surface resistance (R _s), which was observed on a microstrip resonator and may be caused by the edge of the center strip, is reported in this paper for epitaxial high-T _c superconducting (HTSC) thin films. The exhibited behavior is that R _s remains almost unchanged below a certain rf magnetic field H _rf, and then increases abruptly at this field, after which it increases in proportion to H _rf. To explain the behavior, the morphology of the microstrip resonator was examined by atomic force microscopy (AFM), which showed that the edge of the resonator was damaged in some regions because of the acid etching. If the damaged edge is considered as a weakened granular superconductor, the observed R _s behavior could be explained well in terms of the high-frequency critical state model. This implies that the edge condition should be considered in studying the field dependence of R _s when the planar resonator technique is used.     

Implementation of stress-dependent resilient modulus of asphalt-treated base for flexible pavement design

Abstract

Asphalt-treated base (ATB) is a widely used base course material in flexible pavements. Due to its lower binder content and lower quality granular material, resilient modulus (M_R) of ATB exhibits a stress dependent property. While most of the pavement analysis software used in the current routine design tasks is based on Burmister’s multi-layer elastic theory, in which the M_R is the primary fundamental material property, the software cannot truly deal with the stress-dependent M_R of paving material. In this study, M_R model in the Mechanistic Empirical Pavement Design Guide was adopted to accommodate the stress-dependent M_R of ATBs and the model was implemented in pavement mechanistic analyses through Abaqus finite element software and its user programmed subroutine. The comparison of critical pavement responses was made among pavements with three types of ATBs. Pavement responses obtained using the multi-layered elastic theory and based on resilient moduli recommended by Alaska Department of Transportation & Public Facilities, were also included for comparison.     

An economical method for determining the smallest eigenvalues of large linear systems

The general eigenvalue problem A x = λ B x as arising from vibration problems tackled by the finite element method is often solved by the economization method in which the two matrices are reduced to a more manageable size and approximate answers are obtained. This paper analyses the method in a more mathematical way than previous accounts and leads to a definition of the optimum set of variables to be retained during the reduction. An extension to the method is suggested which allows the approximate eigenvalue to be improved and bounds are obtained on the eigenvalues of the full problem before reduction. An estimate is made of the calculation involved in the method and it is concluded, by reference to examples, that the extended method leads to a more efficient algorithm.     

Microwave and Terahertz Surface Resistance of MgB2 Thin Films

The knowledge of the surface resistance R _s of superconducting thin film at microwave and terahertz (THz) regions is significant to design, make and assess superconducting microwave and THz electronic devices. In this paper we reported the R _s of MgB₂ films at microwave and THz measured with sapphire resonator technique and the time-domain THz spectroscopy, respectively. Some interesting results are revealed in the following: (1) A clear correlation is found between R _s and normal-state resistivity right above T _c, ρ₀, i.e., R _s decreases almost linearly with the decrease of ρ₀. (2) A low residual R _s, less than 50 μΩ at 18 GHz is achieved by different deposition techniques. In addition, between 10 and 14 K, MgB₂ has the lowest R _s compared with two other superconductors Nb₃Sn and the high-temperature superconductor YBa₂Cu₃O_7−δ(YBCO). (3) From THz measurement it is found that the R _s of MgB₂ up to around 1 THz is lower than that of copper and YBCO at the temperature below 25 K. (4) The frequency dependence of R _s follows ω ⁿ , where ω is angular frequency, and n is power index. However, n changes from 1.9 at microwave to 1.5 at THz. The above results clearly give the evidences that MgB₂ thin film, compared with other superconductors, is of advantage to make superconducting circuits working in the microwave and THz regions.     

Resonant measurements of surface resistance of high-Tc superconducting films: How good or bad are they?

Errors in measurements of surface resistanceR _s of HTS materials are due to discrepancy between a mathematical model describing physical phenomena and a real measurement environment, finite accuracy of measurements of theQ-factor, and finite accuracy of constants used for calculation ofR _s . In this paper we analyze errors inR _s due to uncertainties in theQ-factor, geometrical coefficients, loss tangent,R _scu, and other factors when a cylindrical copper cavity with an HTS end plate, a stripline resonator, and sapphire rod resonators are used for HTS characterization.     

A new boundary integral equation method of three-dimensional crack analysis

Introducing the mode II and mode III dislocation densities W ₂(y) and W ₃(y) of two variables, a new boundary integral equation method is proposed for the problem of a plane crack of arbitrary shape in a three-dimensional infinite elastic body under arbitrary unsymmetric loads. The fundamental stress solutions for three-dimensional crack analysis and the limiting formulas of stress intensity factors are derived. The problem is reduced to solving three two-dimensional singular boundary integral equations. The analytic solution of the axisymmetric problem of a circular crack under the unsymmetric loads is obtained. Some numerical examples of an elliptical crack or a semielliptical crack are given. The present formulations are of basic significance for further analytic or numerical analysis of three-dimensional crack problems.     

Two-Peak Temperature Dependence of Microwave Surface Impedance of Single-Crystalline YBCO Thin Films: Role of Pairing Symmetry and Defect Structure

Temperature dependencies of microwave surface impedance were measured for c-oriented highly perfect YBCO thin films deposited by off-axis dc magnetron sputtering onto CeO₂-buffered r-cut sapphire substrates. A distinct two-peak structure of R _s(T) and X _s(T) dependencies with peaks at 28–30, K and 50, K has been revealed. The peaks become smeared at higher frequencies or in applied dc magnetic field, while the peak positions remain almost unchanged. The two-peak Z _s(T) behavior is believed to be an intrinsic electron property of extremely perfect quasi-single-crystalline YBCO films. A theoretical model is suggested to explain the observed anomalous Z _s(T) behavior. The model is based on the Boltzman kinetic equation for quasiparticles in layered high-T _c superconductors (HTS) cuprates. It takes into account the supposed s + d wave symmetry of electron pairing and strong energy-dependent relaxation time of quasiparticles, determined mainly by their elastic scattering on extended defects parallel to the c-axis.     

Elastic constants identification of symmetric angle-ply laminates via a two-level optimization approach

A two-level optimization method for elastic constants identification of symmetric angle-ply laminates is presented. Measured axial and lateral strains of two symmetric angle-ply laminates with different fiber angles are used in the proposed method to identify four elastic constants of the composite laminates. In the first-level optimization process, the theoretically and experimentally predicted axial and lateral strains of a [(45°/−45°)₂]_s laminate are used to construct the error function which is a measure of the differences between the experimental and theoretical predictions of the axial and lateral strains. The identification of the material constants is then formulated as a constrained minimization problem in which the best estimates of the shear modulus and Poisson’s ratio of the laminate are determined by making the error function a global minimum. The problem of this level of optimization is then solved using a multi-start global minimization algorithm. In the second-level optimization process, the shear modulus and Poisson’s ratio determined in the previous level of optimization are kept constant while the Young’s moduli of the second angle-ply laminate with fiber angles other than 45° are identified using the same minimization technique that has been used in the previous level. The accuracy of the proposed method are studied by means of a number of numerical examples on the material constants identification of symmetric angle-ply laminates made of different composite materials. Finally, static tensile tests of [(45°/−45°)₂]_s and [(30°/−30°)₂]_s laminates made of Gr/ep composite material are performed to measure the strains of the laminates. The experimental data are then used to identify the elastic constants of the laminates. The excellent results obtained in the experimental investigation have demonstrated the feasibility and applications of the proposed method.