Computational Engineering

This Workshop treated a variety of finite element methods and applications in computational engineering and expanded their mathematical foundation in engineering analysis. Among the 53 participants were mathematicians and engineers with focus on mixed and nonstandard finite element schemes and their applications

ANALYTIC APPROXIMATE SOLUTION FOR NONLINEAR DYNAMICMODELING OF THE ROTATING ELASTIC 2D BEAM WITH A SINGLE CRACK

In this paper, the 2D lateral vibration analysis of a rotating cracked beam as a rotary structure is investigated through the Homotopy perturbation analysis and compared with the numerical Newmark-beta (Nβ) algorithm. The structure and crack are modeled as the Euler-Bernoulli (EB) theory and simple torsional spring, respectively. The nonlinear equations of motion are derived using Galerkin and the Assumed Mode Method (AMM). The system’s stability is analyzed through phase plane and time response for different angular velocities of the base, initial values, external disturbances, crack stiffness, and locations. A comparative study presents simulation results for free (first nonlinear frequency) and forced vibration. It is shown that the proposed semi-analytical approach is beneficial as it provides a benchmark for a more precise analysis and further investigation of cracked rotary structures

Mass transfer in droplets with turbulent internal circulation - Mathematical description

This investigation was carried out to check the accuracy of the eigenvalues, λn, calculated by Wellek and Skelland, using some method other than the Rayleigh-Ritz technique and also, to determine the coefficients, Bn in series solution, which were not calculated before.The eigenvalues were obtained by trial and error procedure using the standard Runge-Kutta method. The coefficients Bn were obtained by using eigenvalues. By using this information the fractional extraction was calculated as a function of droplet contact time. The Wellek-Skelland modification was also solved by finite difference technique to compute fraction extracted as a function of contact time. The eigenvalues, λn, and coefficients, Bn, were also obtained by Hamming\u27s method, and the values agreed with those obtained by the Runge-Kutta method to within 2%. The first eigenvalues obtained by Wellek and Skelland and the values obtained by the author were in agreement to within 5%. The family of curves of Em versus bt, obtained using finite difference method, is in agreement with the physical situation. Therefore, at the higher values of bt ( \u3e 0. 5) results observed using Runge-Kutta method should be used,and at lower values of bt (\u3c 0. 5) the results obtained using finite difference should be considered to be more accurate --Abstract, page ii

Spectral methods for limited area models

1984 Fall.Includes bibliographical references (pages143-150).This study investigates the usefulness of Chebyshev spectral methods in limited area atmospheric modeling. Basic concepts of spectral methods and properties of Chebyshev polynomials are reviewed. Chebyshev spectral methods are illustrated by applying them to the linear advection equation in one dimension. Numerical results demonstrate the high accuracy obtained compared to finite difference methods. The nonlinear shallow water equations on a bounded domain in two dimensions are then considered as a more realistic prototype model. Characteristic boundary conditions based on Reimann invariants are developed, and contrasted with wall conditions and boundary conditions based on the assumption of balanced flow. Chebyshev tau and collocation methods are developed for this model. Results from one-dimensional tests show the superiority of the characteristic conditions in most situations. Results from two-dimensional tests are also presented. Comparison of the tau and collocation methods shows that each has its own advantages and both are practical. Time differencing schemes for Chebyshev spectral methods are studied. The stability condition obtained with explicit time differencing, often thought to be "severe", is shown to be less severe than the corresponding condition for finite difference methods. Numerical results and asymptotic estimates show that time steps may in fact be limited by accuracy rather than stability, in which case simple explicit time differencing is practical and efficient. Two modified explicit schemes are reviewed, and implicit time differencing is also discussed. A Chebyshev spectral method is also used to solve the vertical structure problem associated with vertical normal mode transforms in a hydrostatic atmosphere. Numerical results demonstrate the accuracy of the method, and illustrate the aliasing which can occur unless the vertical levels at which data is supplied are carefully chosen. Vertical transforms of observed forcings of tropical wind and mass fields are presented. The results of this study indicate that Chebyshev spectral methods are a practical alternative to finite difference methods for limited area modeling, especially when high accuracy is desired. Spectral methods require less storage than finite difference methods, are more efficient when high enough accuracy is desired, and are at least as easy to program.Supported by the National Science Foundation - ATM-8207563.Supported by the Office of Naval Research - N00014-84-C-0591.Acknowledgment to the National Center for Atmospheric Research, sponsored by the National Science Foundation, for computer time

On the Influence of Multiplication Operators on the Ill-posedness of Inverse Problems: Zum Einfluss von Multiplikationsoperatoren auf die Inkorrektheit Inverser Probleme

In this thesis we deal with the degree of ill-posedness of linear operator equations in Hilbert spaces, where the operator may be decomposed into a compact linear integral operator with a well-known decay rate of singular values and a multiplication operator. This case occurs for example for nonlinear operator equations, where the local degree of ill-posedness is investigated via the Frechet derivative. If the multiplier function has got zeroes, the determination of the local degree of ill-posedness is not trivial. We are going to investigate this situation, provide analytical tools as well as their limitations. By using several numerical approaches for computing the singular values of the operator we find that the degree of ill-posedness does not change through those multiplication operators. We even provide a conjecture, verified by several numerical studies, how these multiplication operators influence the singular values of the operator equation. Finally we analyze the influence of those multiplication operators on the opportunities of Tikhonov regularization and corresponding convergence rates. In this context we also provide a short summary on the relationship between nonlinear problems and their linearizations.Diese Arbeit beschaeftigt sich mit dem Grad der Inkorrektheit linearer Operatorgleichungen in Hilbertraeumen, die sich als Komposition eines vollstetigen linearen Integraloperators mit bekannter Abklingrate der Singulaerwerte und eines Multiplikationsoperators darstellen lassen. Dieser Fall tritt beispielsweise bei nichtlinearen Operatorgleichungen auf, wobei der lokale Inkorrektheitsgrad ueber die Frechetableitung bestimmt wird. Falls die Multiplikatorfunktion Nullstellen hat, so ist die Bestimmung des lokalen Grades der Inkorrektheit nicht einfach. Moeglichkeiten und Grenzen der Analysis fuer diese Situation werden betrachtet. Unterschiedliche numerische Ansaetze fuer die Bestimmung der Singulaerwerte liefern, dass der Grad der Inkorrektheit durch die Multiplikationsoperatoren nicht veraendert wird. Es wird sogar ein Zusammenhang angegeben, wie Multiplikationsoperatoren die Singulaerwerte beeinflussen. Schliesslich werden Moeglichkeiten der Tikhonov-Regularisierung unter Einfluss der Multiplikationsoperatoren untersucht. In diesem Zusammenhang wird auch eine kurze Zusammenfassung zur Beziehung von nichtlinearen Problemen und ihren Linearisierungen gegeben