Analytic solution to the problem of the temperature jump in a gas with rotational degrees of freedom

An exact solution is obtained for the first time for the problem of the temperature jump in a gas with allowance for internal (rotational) degrees of freedom. The treatment is based on a model collision integral proposed by the authors. The problem reduces to the solution of a boundary-value problem for a linear vector transport equation with matrix kernel. Separation of the variable leads to a characteristic equation for which eigenvectors are found in a space of generalized functions and the eigenvalue spectrum is investigated. An expansion of the solution to the problem with respect to eigenvectors of the continuous and discrete spectra is established. On the basis of the conditions of solvability of the vector Riemann-Hilbert boundary-value problem which arises in the process of the proof, an exact (in closed form) expression is obtained for the temperature jump.Moscow Pedagogical University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 95, No. 3, pp. 530–540, June, 1993.     

Eigenwaves in a lossy metal-dielectric waveguide

ABSTRACT

The problem of normal waves in a closed (shielded) regular waveguide of arbitrary cross-section is considered. This problem is reduced to the boundary eigenvalue problem for longitudinal components of electromagnetic field in Sobolev spaces. To find the solution, we use the variational formulation of the problem. The variational problem is reduced to study an operator-function. Discreteness of the spectrum is proved and distribution of the characteristic numbers of the operator-function on the complex plane is found. We also consider properties of system of eigenvectors and associated vectors of the operator-function. Double completeness of system of eigenvectors and associated vectors with a finite defect is established.     

Discrete Sturm-Liouville Problems With Parameter in the Boundary Conditions

This paper deals with discrete second order Sturm-Liouville problems in which the parameter that is part of the Sturm-Liouville difference equation also appears linearly in the boundary conditions. An appropriate Green's formula is developed for this problem, which leads to the fact that the eigenvalues are simple, and that they are real under appropriate restrictions. A boundary value problem can be expressed by a system of equations, and finding solutions to a boundary value problem is equivalent to finding the eigenvalues and eigenvectors of the coefficient matrix of a related linear system. Thus, the behavior of eigenvalues and eigenvectors is investigated using techniques in linear algebra, and a linear-algebraic proof is given that the eigenvalues are distinct under appropriate restrictions. The operator is extended to a self-adjoint operator and an expansion theorem is proved.     

Using eigenstructure of the Hessian to reduce the dimension of the intensity modulated radiation therapy optimization problem

Optimization is of vital importance when performing intensity modulated radiation therapy to treat cancer tumors. The optimization problem is typically large-scale with a nonlinear objective function and bounds on the variables, and we solve it using a quasi-Newton sequential quadratic programming method. This study investigates the effect on the optimal solution, and hence treatment outcome, when solving an approximate optimization problem of lower dimension. Through a spectral decompostion, eigenvectors and eigenvalues of an approximation to the Hessian are computed. An approximate optimization problem of reduced dimension is formulated by introducing eigenvector weights as optimization parameters, where only eigenvectors corresponding to large eigenvalues are included. The approach is evaluated on a clinical prostate case. Compared to bixel weight optimization, eigenvector weight optimization with few parameters results in faster initial decline in the objective function, but with inferior final solution. Another approach, which combines eigenvector weights and bixel weights as variables, gives lower final objective values than what bixel weight optimization does. However, this advantage comes at the expense of the pre-computational time for the spectral decomposition. A preliminary version of this paper was presented at the AAPM 46th annual meeting, held July 25–29, 2004 in Pittsburgh, PA.     

The compression of a thin strip of material in a superplasticity state

The viscoplastic flow of a thin strip of material in a superplasticity state between rigid, converging parallel planes (an analogue of Prandtl's problem) is investigated. An analytical quadrature solution of the problem is constructed, asymptotically precise in the same sense as Prandtl's solution. Special cases are considered where the solution (including an approximate solution) is written out fully. The effects of superplasticity are determined.     

Quadratic model updating with no spill-over and incomplete measured data: Existence and computation of solution

Quadratic finite element model updating problem (QFEMUP), to be studied in this paper, is concerned with updating a symmetric nonsingular quadratic pencil in such a way that, a small set of measured eigenvalues and eigenvectors is reproduced by the updated model. If in addition, the updated model preserves the large number of unupdated eigenpairs of the original model, the model is said to be updated with no spill-over. QFEMUP is, in general, a difficult and computationally challenging problem due to the practical constraint that only a very small number of eigenvalues and eigenvectors of the associated quadratic eigenvalue problem are available from computation or measurement. Additionally, for practical effectiveness, engineering concerns such as nonorthogonality and incompleteness of the measured eigenvectors must be considered. Most of the existing methods, including those used in industrial settings, deal with updating a linear model only, ignoring damping. Only in the last few years a small number of papers been published on the quadratic model updating; several of the above issues have been dealt with both from theoretical and computational point of views. However, mathematical criterion for existence of solution has not been fully developed. In this paper, we first (i) prove a set of necessary and sufficient conditions for the existence of a solution of the no spill-over QFEMUP, then (ii) present a parametric representation of the solution, assuming a solution exists and finally, (iii) propose an algorithm for QFEMUP with no spill-over and incomplete measured eigenvectors. Interestingly, it is shown that the parametric representation can be constructed with the knowledge of only the few eigenvalues and eigenvectors that are to be updated and the corresponding measured eigenvalues and eigenvectors—complete knowledge of eigenvalues and eigenvectors of the original pencil is not needed, which makes the solution readily applicable to real-life structures.     

Damage identification in a concrete dam by fitting measured modal parameters

We propose a method, based on an inverse problem, to obtain numerically the material parameters that characterize the elasticity tensor of a body with linear elastic behavior, using accurate measurements of the first modal parameters, namely the natural frequencies and the modes of vibration (the eigenfrequencies and the eigenvectors). Appropriate functionals are defined, whose minimum points correspond to the unknown material parameters. To obtain these minimum points a highly nonlinear parametric optimization problem is solved. Its resolution involves specific mathematical tools like the derivative of the eigenvalues and eigenvectors with respect to the material parameters, the adjoint method, and gradient methods for the minimization of the functional. An application is presented, which considers a cracked dam in which is assumed the presence of transversely isotropic material in the cracked zone. The material parameters of the transversely isotropic material are obtained by minimizing the distance between the modal parameters (eigenfrequencies and eigenvectors) of a numerical model of the dam and the observed modal parameters physically measured in the dam. The algorithm is implemented in a C++ home made code with the aid of open-source libraries for scientific computation.     

An extremal property of the eigenvalue of irreducible matrices in idempotent algebra and solution of the Rawls location problem

An extremal property of the eigenvalue of an irreducible matrix in idempotent algebra is studied. It is shown that this value is the minimum value of some functional defined using this matrix on the set of vectors with nonzero components. The minimax problem of location of a single facility (the Rawls problem) on a plane with rectilinear distance is considered. For this problem, we give the corresponding representation in terms of idempotent algebra and suggest a new algebraic solution, which is based on the results of investigation of the extremal property of eigenvalue and reduces to finding the eigenvalue and eigenvectors of a certain matrix.     

Backward error of polynomial eigenvalue problems solved by linearization

It is commonplace in many application domains to utilize polynomial eigenvalue problems to model the behaviour of physical systems. Many techniques exist to compute solutions of these polynomial eigenvalue problems. One of the most frequently used techniques is linearization, in which the polynomial eigenvalue problem is turned into an equivalent linear eigenvalue problem with the same eigenvalues, and with easily recoverable eigenvectors. The eigenvalues and eigenvectors of the linearization are usually computed using a backward stable solver such as the QZ algorithm. Such backward stable algorithms ensure that the computed eigenvalues and eigenvectors of the linearization are exactly those of a nearby linear pencil, where the perturbations are bounded in terms of the machine precision and the norms of the matrices defining the linearization. Although we have solved a nearby linear eigenvalue problem, we are not certain that our computed solution is in fact the exact solution of a nearby polynomial eigenvalue problem. Here, we perform a backward error analysis for the solution of a specific linearization for polynomials expressed in the monomial basis. We use a suitable one-sided factorization of the linearization that allows us to map generic perturbations of the linearization onto structured perturbations of the polynomial coefficients. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The free-parameter solution of the convection equations in a vertical cylinder with a volume heat source

An exact solution of the free-convection equations is constructed in the Oberbeck–Boussinesq approximation, describing the flow of a viscous heat-conducting fluid in a vertical cylinder of large radius when heated by radiation. The initial problem is reduced to an operator equation with an extremely non-linear operator, satisfying Schauder's theorem in C[0,1]. An iteration procedure is proposed for determining the independent parameter, that occurs in the solution, which enables three different values to be obtained and, correspondingly, three classes of solution of the initial problem. The linear stability of all the solutions obtained is investigated and it is shown that, for chosen values of the problem parameters, the most dangerous one is the plane wave mode and two instability mechanisms are present. The flow structure and the type of instability depend considerably on the values of the free parameter.     

A Riemann solver for RANS

An exact expression for a system of both eigenvalues and right/left eigenvectors of a Jacobian matrix for a convective two-equation differential closure RANS operator split along a curvilinear coordinate is derived. It is shown by examples of numerical modeling of supersonic flows over a flat plate and a compression corner with separation that application of the exact system of eigenvalues and eigenvectors to the Roe approach for approximate solution of the Riemann problem gives rise to an increase in the convergence rate, better stability and higher accuracy of a steady-state solution in comparison with those in the case of an approximate system.     

Eigenvectors of certain matrices

An algorithm is given for calculation of eigenvalues and eigenvectors of centrosymmetric and some related matrices, and some desirable properties of the algorithm are proved. Centrosymmetric matrices are characterized by a symmetry property of their eigenvectors and this result is used to establish a property of certain methods for the numerical solution of differential equations.     

The role of cross terms in the flow law fora gas condensate mixture in a porous medium near a well

The problem of the steady flow of a gas condensate mixture in a porous medium near a well is considered. Capillary forces and cross terms are taken into account in the flow law. It is shown that the problem splits into the purely physicochemical problem of capillary condensation and the problem of determining of the pressure field in the phases. The latter problem can have infinitely many solutions; the solution which is of the greatest interest in practice is separated out. An approximate solution in analytical form is found in the case of a small ratio of the gas and condensate viscosities.     

Magnetic Eigenmaps for the visualization of directed networks

We propose a framework for the visualization of directed networks relying on the eigenfunctions of the magnetic Laplacian, called here Magnetic Eigenmaps. The magnetic Laplacian is a complex deformation of the well-known combinatorial Laplacian. Features such as density of links and directionality patterns are revealed by plotting the phases of the first magnetic eigenvectors. An interpretation of the magnetic eigenvectors is given in connection with the angular synchronization problem. Illustrations of our method are given for both artificial and real networks.     

点时滞系统的反馈镇定

求解特征矩阵是镇定时滞系统的关键问题，本文给出了系统的特征根的代数重复度与几何重复度均为一般值的情况下特征矩阵的求法，即把它归结为求一组线性代数方程的问题，并得到了该方程组有组及对应于同一特征值的解向量组线性独立的充分条件。本文还提出了一种算法来处理系统对应于不同特征值的左行征向量线性相关情况下系统的镇定问题最后，举例说明了设计步骤。     

Behavior of the formal solution to a mixed problem for the wave equation

The behavior of the formal solution, obtained by the Fourier method, to a mixed problem for the wave equation with arbitrary two-point boundary conditions and the initial condition φ(х) (for zero initial velocity) with weaker requirements than those for the classical solution is analyzed. An approach based on the Cauchy–Poincare technique, consisting in the contour integration of the resolvent of the operator generated by the corresponding spectral problem, is used. Conditions giving the solution to the mixed problem when the wave equation is satisfied only almost everywhere are found. When φ(x) is an arbitrary function from L₂[0, 1], the formal solution converges almost everywhere and is a generalized solution to the mixed problem.     

Inverse Problem for a Mathematical Model of Adsorption Dynamics

A mixed problem is considered for a system of partial differential equations modeling the process of adsorption dynamics. An existence and uniqueness theorem is proved for this problem, and the solution properties are investigated. The inverse problem is posed, involving the determination of the system coefficient given additional information about the solution. A uniqueness theorem is proved for the solution of the inverse problem.__________Translated from Prikladnaya Matematika i Informatika, No. 16, pp. 5 – 14, 2004.     

Extension of the solution of nonlinear equations in the neighborhood of a bifurcation point

Using the method of continuous extension with respect to a parameter we develop a method of constructing the load trajectory of a structure having both limit points and bifurcation points. The method is applicable for the systems of nonlinear algebraic equations that describe the family of extremals that minimize the value of the total potential strain energy of the structure, and makes it possible to find all the branches of the load trajectory emanating from a bifurcation point and extend the solution along any of them. The method is based on the fact that the eigenvectors of the augmented Jacobian of the system of equations in the extended space of variables that correspond to zero eigenvalues on the main branch of the load trajectory are bifurcation vectors and form the active subspace of solutions of the equations of the extension. Meanwhile the other eigenvectors form the passive subspace that contains the extension vector with respect to the main branch of the load. As a result the entire process of computing the extension vector of the solution at any point of the load trajectory reduces to determining the eigenvectors of the augmented Jacobian of the original system of nonlinear algebraic equations, identifying them according as they belong to the active or passive subspace, and forming the extension vector of the solution using them and analytic relations Translated fromMatematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 1, 1998, pp. 35–46.     

Solving the problem of contact equilibrium of a punch and an elastic body

The variational problem of contact equilibrium of a punch and an elastic body is considered. An equivalent formulation of the problem is given in variational inequality form. Existence and uniqueness of the solution is investigated in a particular case. A penalty method is proposed for approximate solution of the problem.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 56, pp. 97–103, 1985.     

Convergence of the solution for a domain decomposed,heterogeneously modeled flow past a concave profile problem

The free streamline problem investigated is that of fluid flow past a symmetric truncated concave‐shaped profile between walls. An open wake or cavity is formed behind the profile. Conformal mapping techniques are used to solve this problem. The problem formulated in the hodograph plane is decomposed into two nonoverlapping domains. Heterogeneous modeling is then used to describe the problems, i.e., a different governing differential equation in each domain. In one of these domains, a Baiocchi‐type transformation is used to obtain a fixed domain formulation for the part of the transformed problem containing an unknown boundary. In the other domain, the Baiocchi‐type transformation is extended across the boundary between the two domains, thus yielding a different problem formulation. This also assures that the dependent variables and their normal derivatives are continuous along this common boundary. The numerical solution scheme, a successive over‐relaxation approach, is applied over the whole problem domain with the use of a projection‐operation over only the fixed domain formulated part. Numerical results are obtained for the case of a truncated circular profile. These results are found to be in good agreement with another published result. The existence and uniqueness of the solution to the problem as a variational inequality is shown, and the convergence of the numerical solution using a domain decomposition method scheme is demonstrated by assuming some convergence property on the common interface of the two subdomains. © 2000 John Wiley & Sons, Inc. Numeer Methods Partial Differential Eq 16: 459–479, 2000