Unknown Boundary Values Research Articles

Improvement of Accuracy of Inverse Analysis for Stress Separation in Thermoelastic Stress Analysis.

This paper considers determination of individual stress components from the sum of the principal stresses obtained experimentally by the thermoelastic stress analysis. The stress separation problem is divided into two parts : (1) an inverse problem to estimate the unknown boundary values from the knowledge of the sum of the principal atresses inside the analysis region, snd (2) a forward problem to compute the stress components inside the analysis region based on the estimated boundary values. These two problems can be formulated and solved by the BEM. As the inverse problem is often ill-posed, two techniques are adopted so as to attain an accurate result. One is pre-processing of experimental data, that is filtering based on the compatibility equation. The other is regularization of the inverse problem by Tikhonov's method with Hansen's L-curve method. The effectiveness of developed method is verified by applying it to an experimental data. The effect of configuration of the analysis region on the accuracy of stress separation is also discussed.

Boundary Element Inverse Analysis for Stress Separation in Thermoelastic Stress Analysis.

Thermoelastic stress analysis(TSA) is a technique for measuring stresses through temperature changes induced by the thermoelastic effect of elastic bodies. Unfortunately, TSA can only provide the sum of the principal stresses. In this paper, the stress separation problem to determine individual stress components from the sum of the principal stresses is considered. The problem is divided into two parts:(1)an inverse problem to estimate unknown boundary values from the sum of the principal stresses inside the body, and(2)a direct problem to determine the stress components inside the body based on the estimated boundary values. This two-step approach is formulated and solved by BEM. It is found that the combination of Tikhonov regularization with supplementary data and Hansen's L-curve method is effective for attaining an accurate result of stress separation.

Improvement of Accuracy of Inverse Analysis for Stress Separation in Thermoelastic Stress Analysis.

This paper considers determination of individual stress components from the sum of the principal stresses obtained experimentally by the thermoelastic stress analysis. The stress separation problem is devided into two parts : (1) an inverse problem to estimate the unknown boundary values from the knowledge of the sum of the principal stresses inside the analysis region, and (2) a forward problem to compute the stress components inside the analysis region based on the estimated boundary values. These two problems can be formulated and solved by the BEM. As the inverse problem is often ill-posed, two techniques are adopted so as to attain an accurate result. One is preprocessing of experimental data, that is filtering based on the compatibility equation. The other is regularization of the inverse problem by Tikhonov's method with Hansen's L-curve method. The effectiveness of developed method is verified by applying it to an experimental data. The effect of configuration of the analysis region on the accuracy of stress separation is also discussed.

Vibrations of composite circular cylindrical vessels

Free vibrations are analyzed for a laminated composite circular cylindrical vessel consisting of a laminated composite circular cylindrical shell and laminated composite circular plate lids with a hole at both ends. In the analysis, by using the exact solutions of the equations of motion for the circular cylindrical shell and the circular plate, the Lagrangian of the vessel is expressed in a quadratic form of the unknown boundary values. The frequency equations are obtained by minimizing the Lagrangian with respect to the unknown boundary values. Numerical studies are made for vessels with symmetric cross-ply laminates of unidirectional graphite fiber reinforced epoxy. The natural frequencies, the mode shapes and the bending moment distributions are obtained and their characteristics are investigated.

Determination of Open Boundary Conditions with an Optimization Method

The optimization method proposed in this paper is for determining open boundary conditions from interior observations. Unknown open boundary conditions are represented by an open boundary parameter vector (B), while known interior observational values are used to form an observation vector (O). For a hypothetical B* (generally taken as the zero vector for the first time step and as the optimally determined B at the previous time step afterward), the numerical ocean model is integrated to obtain solutions (S*) at interior observation points. The root-mean-square difference between S* and O might not be minimal. The authors change B* with different increments dB. Optimization is used to get the best B by minimizing the error between O and S. The proposed optimization method can be easily incorporated into any ocean models, whether linear or nonlinear, reversible or irreversible, etc. Applying this method to a primitive equation model with turbulent mixing processes such as the Princeton Ocean Model (POM), an important procedure is to smooth the open boundary parameter vector. If smoothing is not used, POM can only be integrated within a finite period (45 days in this case). If smoothing is used, the model is computationally stable. Furthermore, this optimization method performed well when random noise was added to the ‘‘observational’’ points. This indicates that realtime data can be used to inverse the unknown open boundary values.

Inverse Analysis Related to Stress Separation in Thermoelastic Stress Analysis

Thermoelastic stress analysis(TSA) is a technique for measuring stresses through temperature changes induced by the thermoelastic effect of elastic bodies. Since the TSA can only provide the sum of the principal stresses, stress separation is required to obtain individual stress components. This paper deals with stress separation of TSA data. The problem is divided into two: (1) an inverse problem to estimate the unknown boundary values from the sum of the principal stresses inside the body, and (2) a direct problem to derive the stress components inside the body on the basis of the estimated boundary values. This two-step method can be applied to a wide range of practical problems if both the inverse and direct problems are solved by FEM or BEM. In this study, the problem is formulated by BEM. It is shown that careful treatment of the inverse problem is essential for attaining an accurate result of the stress separation. The combination of the truncated singular value decomposition and Hansen's L-curve method is found to be effective for the purpose.

On the Inverse Problem for the Ekman Equation

We consider inverse problems for the partial differential equation Δu - 2h -1 (⊇h, ⊇u) = g. We assume that the unknown function u - the stream function for a class of flow problems - is constant on certain parts of the boundary ∂Ω while at the remaining parts no boundary condition is imposed. Instead, values of the gradient of u (velocities) are prescribed at a finite set M of inner points (measuring points). We look for the minimizer of a functional being a combination of the squared errors at the measuring points and - for regularization - a squared H 1/2 -norm ofthe (unknown) boundary values. We present numerical solutions to the problem. Especially, the dependence on the regularization parameter μ is discussed.

Inverse Analysis Related to Stress Separation in Thermoelastic Stress Analysis.

This paper deals with stress separation in thermoelastic stress analysis. The problem of stress separation can be divided into two : (1) an inverse problem to estimate the unknown boundary values from the sum of the principal stresses inside the body, and (2) a direct problem to compute the stress components inside the body based on the estimated boundary values. This two-step method can be applied to a wide range of practical problems if both the inverse and direct problems are solved by FEM or BEM. In this study, the problem is formulated by BEM. It is shown that careful treatment of the inverse problem is essential for attaining an accurate result of the stress separation. The combination of the truncated singular value decomposition and Hansen's L-curve method is found to be effective for that purpose.

Stochastic boundary elements for two-dimensional potential flow in non-homogeneous media

A stochastic boundary element formulation is presented for the treatment of two-dimensional problems of steady-state potential flow in non-homogeneous media that involve a random operator in the governing differential equation. The randomness is introduced through the material parameter of the domain which is described as a non-homogeneous random field. The random field is discretized into a set of correlated random variables and a perturbation is applied to the differential equation of the problem. This leads to differential equations for the unknown potential and its first- and second-order derivatives, respectively, evaluated at the mathematical expectations of the random variables resulting from the discretization of the random field. An approximate method is applied for the solution of these equations by expressing the potential and its derivatives as a sum of functions of descending order. These solutions are introduced into the differential equations and upon equating similar order terms, a sequence of Poisson's equations is obtained for each order. A transformation of the correlated random variables into an uncorrelated set is performed to reduce the number of numerical operations by retaining a small number of transformed random variables. The resulting equations are solved using the boundary element method to obtain the unknown boundary values of the potentials and their respective first- and second-order derivatives which are then used to compute the desired response statistics. Quadratic, conforming boundary elements are used in the boundary integration and four-node quadrilateral cells are used in the domain integration. Strongly singular terms of the boundary element matrices are obtained indirectly by applying a state of uniform unit potential over the entire contour of the object. The singular domain integrals are calculated analytically. Direct solution techniques are used to calculate the response variables and their derivatives, respectively. A number of example problems are presented and the results are compared with those obtained from Monte Carlo simulations. A good agreement of the results is observed.

Considerations of Elastodynamic Inverse Analysis Using Extended Kalman Filter Algorithm.

This paper is concerned with elastodynamic inverse analysis in which unknown defects, cracks and/or boundary conditions should be identified from displacement responses on the boundary of an elastic body. In the previous paper, the authors proposed the inverse analysis method for identification of unknown parameters using the extended Kalman filter algorithm and the boundary element analysis for steady-state elastodynamics. In this study, computational aspects of the inverse analysis method are investigated with respect to the influence of errors in the measured data and to arrangement of the measuring points. Furthermore, computational efficiency and accuracy are compared between the inverse analyses using the extended Kalman filter algorithm and the optimization method. Numerical simulation is carried out for some example problems to identify an unknown defect and unknown boundary values as a parameter identification problem.

An analysis of the scalar Helmholtz equation using the integral equation method

The scalar Helmholtz equations are investigated by using the integral equation method (IEM). In the IEM analysis, the fundamental solution of the Laplace equation is used as a weighting function. Two IEM formulations are obtained: one is a standard formulation and the other is obtained from an elimination of the unknown boundary value. The accuracy and computational time of the IEM are compared with those of the finite element method in two dimensional scalar Helmholtz problems. The analysis of a resonant cavity is reduced to a simple eigenvalue problem. Resonant frequencies of the IEM agree well with those of the finite difference method. Usefulness of the IEM is confirmed through the analyses of the scalar Helmholtz equations. >

Free vibrations of a vessel consisting of circular plates and a shell of revolution having varying meridional curvature

An analytical solution procedure is presented for the free vibration of vessels consisting of a shell of revolution having varying meridional curvature and circular plate lids. The Lagrangian of vibration of the combined system is obtained in quadratic forms of boundary values. The frequency equation and the relations among the boundary values are obtained from minimizing conditions of the Lagrangian with respect to the unknown boundary values. The natural frequencies and the mode shapes of vessels having elliptical and hyperbolical meridians have been obtained by carrying out numerical calculations. Effects of various parameters upon natural frequencies and mode shapes are illustrated in discussions of numerical results.

An Integral Equation Method for a Problem with Mixed Boundary Conditions

Consider the problem of finding a complex function $f = u + iv$, which is holomorphic in a given domain, with the real part u taking prescribed values on one part of the boundary, and the imaginary part v taking prescribed values on the remainder of the boundary. (This is essentially equivalent to solving Laplace’s equation subject to mixed Dirichlet and Neumann boundary conditions.) By virtue of the Cauchy integral formula, the unknown boundary values of u and v satisfy a $2 \times 2$ system of singular integral equations, which can be solved in a certain class of weighted $L_p $ spaces, by applying a result of I. Gohberg and N. Krupnik. The values of both u and v are then known over the whole of the boundary, and so f can be computed using the Cauchy integral formula.

A new method of multisegment integration for the nonlinear analysis of shells of revolution

A new method of multisegment integration has been developed here for solving boundary value problems of a system of nonlinear ordinary differential equations as encountered in the large deflection analysis of the axisymmetric deformations of the shells of revolution. Multisegment integration is used to solve those boundary value problems of ordinary differential equations which cannot be solved by direct integration, because the latter loses its accuracy in the process of subtraction of almost equal numbers in evaluating the unknown boundary values. The method developed here involves much less computational work than the multisegment integration developed by Kalnins and Lestingi. The new method has been applied to a number of problems to ascertain its soundness and accuracy.

Solution of elastic-plastic shallow shell problems by the boundary element method

The boundary integral equations for elasto-plastic problems of shallow shells are established by using the fundamental solutions of shallow shells derived previously. The strains and stress-resultants in the plastic region are used as unknown variables. The simultaneous nonlinear equations of these variables and unknown boundary values are established and solved by direct iteration method.

BOTTOM TURBULENT BOUNDARY LAYER IN WAVE-CURRENT CO-EXISTING SYSTEMS

This study presents a new mathematical method calculating the water particle velocity in the wave-current co-existing systems. A boundary layer thickness 5,,, in the co-existing system is expected to be variable with the water particle velocity ratio of wave component to current component. In this method, the boundary layer equation is solved as a free boundary problem by treating 6,,, as an unknown boundary value. Several characteristics of the turbulent boundary layer such as the friction factor, friction velocity, boundary layer thickness, etc. are calculated by this method and the effect of the wave-current velocity ratio on them is discussed. Furthermore, the velocity reduction of the current due to wave superimposing is investigated. In addition, near-bottom velocities are measured by a laser-doppler velocimeter in the pure current, the pure wave and the wave-current co-existing fields. These results are compared with calculated ones by this mathematical method.

Vibrations of a Combined System of Circular Plates and a Shell of Revolution

In this paper, we analyze the asymmetric vibrations of a combined system of a barrel-like shell of revolution and circular plate lids. The Lagrangian of the combined system is expressed in quadratic forms of boundary values, and the frequency equations can be obtained from the minimum conditions of that Lagrangian with respect to unknown boundary values. Effects of the circular plate lids, the ratio of thickness of the circular plate to that of the shell of revolution, circumferential wave numbers upon natural frequencies of a combined system are clarified in a discussion of numerical results. Transfer phenomena of the mode shapes are also investigated.

On the solution of the linear‐quadratic optimal control problem by extended invariant imbedding

AbstractThe solution of the linear‐quadratic output regulator problem by Pontryagin's maximum principle results in an unstable linear boundary value problem. The method of invariant imbedding, which requires the integration of a matrix Riccati differential equation, solves certain boundary value problems in a stable manner. In order to make this well known method applicable, an extension algorithm is defined, which maps the boundary value problem into a problem of double dimension. The resulting algorithm of extended invariant imbedding does not depend on the boundary values. So the matrix Riccati equation has to be integrated only once ‘offline’: The unknown boundary values, the state, and the control are calculated by solution of systems of linear equations (‘online’). So, if the boundary values change, only systems of linear equations have to be solved once more. The algorithm has two variants with usually contrary stability behaviour.

An efficient numerical technique for the solution of a nonlinear capillary wave problem

Buneman's block cyclic reduction method has been modified to solve a capillary wave problem which is characterized by a nonlinear boundary condition on the free surface. This technique requires an iterative solution of only the unknown boundary values along the free streamline. Once the free surface values are known, the remaining unknowns are determined directly. Comparisons of the numerical solutions with existing exact solutions demonstrate the accuracy and efficiency of this method for handling the nonlinear boundary condition. Some generalizations of the method have also been discussed.

Vibrations of a Cylindrical Shell with Variable Thickness Capped by a Circular Plate

In this paper, the vibrations of a combined system of a circular cylindrical shell with variable thickness of which one end is clamped and the other is connected to a circular plate with variable thickness are analyzed by means of the improved thick plate and shell theories. The Lagrangian of the combined system is expressed by quadratic forms of boundary values, and the frequency equations can be obtained from the minimum conditions of that Lagrangian with respect to unknown boundary values. Effects of a circular plate and some parameters such as variations of thickness upon natural frequencies of the combined system are clarified in discussions of numerical results. Transfer phenomena of the mode shapes are also investigated.