Accurate computation of the high dimensional diffraction potential over hyper-rectangles

We propose a fast method for high order approximation of potentials of the Helmholtz type operator Delta+kappa^2 over hyper-rectangles in R^n. By using the basis functions introduced in the theory of approximate approximations, the cubature of a potential is reduced to the quadrature of one-dimensional integrals with separable integrands. Then a separated representation of the density, combined with a suitable quadrature rule, leads to a tensor product representation of the integral operator. Numerical tests show that these formulas are accurate and provide approximations of order 6 up to dimension 100 and kappa^2=100

Highly Localized RBF Lagrange Functions for Finite Difference Methods on Spheres

The aim of this paper is to show how rapidly decaying RBF Lagrange functions on the spheres can be used to create effective, stable finite difference methods based on radial basis functions (RBF-FD). For certain classes of PDEs this approach leads to precise convergence estimates for stencils which grow moderately with increasing discretization fineness

Particular Solutions of Products of Helmholtz-Type Equations Using the Matern Function

We derive closed-form particular solutions for Helmholtz-type partial differential equations. These are derived explicitly using the Matern basis functions. The derivation of such particular solutions is further extended to the cases of products of Helmholtz-type operators in two and three dimensions. The main idea of the paper is to link the derivation of the particular solutions to the known fundamental solutions of certain differential operators. The newly derived particular solutions are used, in the context of the method of particular solutions, to solve boundary value problems governed by a certain class of products of Helmholtz-type equations. The leave-one-out cross validation (LOOCV) algorithm is employed to select an appropriate shape parameter for the Matern basis functions. Three numerical examples are given to demonstrate the effectiveness of the proposed method

Нестаціонарні теплові процеси в анізотропних твердих тілах

Дисертацію присвячено дослідженню теплових процесів в анізотропних твердих тілах за допомогою безсіткового методу розв’язання тривимірних задач нестаціонарної теплопровідності. Серед великого розмаїття задач математичної фізики, які нині успішно вирішуються, особливу роль займають задачі теплопровідності в анізотропних матеріалах. Насамперед це пов’язано з активним використанням анізотропних матеріалів при виготовленні великої кількості сучасних приладів та пристроїв, деталей конструкцій та машин – наприклад, трансформаторів із сердечниками з текстурованої сталі (в електротехніці), лопаток газотурбінних двигунів із жароміцних нікелевих сплавів з монокристалічною структурою (в авіації), п’єзоперетворювачів, електрооптичних модуляторів та рідкокристалічних індикаторів (в електронному приладобудуванні). Сучасні анізотропні матеріали зі складною структурою (наприклад, композитні матеріали, багатошарові матеріали, покриття, нанесені на підкладки) все частіше використовуються в новітніх інженерних розробках, а також в якості конструкційних матеріалів.У різних технологічних процесах і пристроях дані матеріали піддаються тепловому впливу, внаслідок чого в них відбуваються фізико-хімічні явища, зокрема зміна геометричних розмірів. Неконтрольоване теплове розширення конструкційних матеріалів може призвести до погіршення експлуатаційних характеристик пристрою, а також до аварійних ситуацій. Тому при створенні та використанні таких матеріалів необхідно враховувати анізотропію їх теплофізичних властивостей, а також досліджувати теплові процеси, які в них протікають. The dissertation deals with the study of thermal processes in anisotropic solids by meshless method for solving three-dimensional non-stationary heat conduction problems. Heat conduction problems in anisotropic solids play a significant role among the wide variety of problems of mathematical physics which are currently being successfully solved. First of all, it is associated with the active use of anisotropic materials in the manufacture of a large number of modern instruments and devices, structural parts and machines. For example, transformers with textured steel cores (in electrical engineering), gas-turbine engine blades of heat-resistant nickel alloys with a single-crystal structure (in aviation), piezoelectric transducers, electro-optic modulators and liquid crystal indicators (in electronic instrument engineering). Modern anisotropic materials with a complex structure (composite materials, multilayered materials, coatings on substrates, etc.) are increasingly used in advanced engineering designs and as structural materials. In various technological processes and devices, these materials are exposed to thermal effects, resulting in physical and chemical phenomena, including changes in geometric parameters. Uncontrolled thermal expansion of structural materials may lead to device performance degradation, as well as to emergency situations. Therefore, when creating and using such materials, it is necessary to take into account the anisotropy of their thermophysical properties, as well as to study the thermal processes occurring in them

A boundary integral equation method for numerical solution of parabolic and hyperbolic Cauchy problems

We present a unified boundary integral approach for the stable numerical solution of the ill-posed Cauchy problem for the heat and wave equation. The method is based on a transformation in time (semi-discretisation) using either the method of Rothe or the Laguerre transform, to generate a Cauchy problem for a sequence of inhomogenous elliptic equations; the total entity of sequences is termed an elliptic system. For this stationary system, following a recent integral approach for the Cauchy problem for the Laplace equation, the solution is represented as a sequence of single-layer potentials invoking what is known as a fundamental sequence of the elliptic system thereby avoiding the use of volume potentials and domain discretisation. Matching the given data, a system of boundary integral equations is obtained for finding a sequence of layer densities. Full discretisation is obtained via a Nyström method together with the use of Tikhonov regularization for the obtained linear systems. Numerical results are included both for the heat and wave equation confirming the practical usefulness, in terms of accuracy and resourceful use of computational effort, of the proposed approach

Material State Awareness for Composites Part II: Precursor Damage Analysis and Quantification of Degraded Material Properties Using Quantitative Ultrasonic Image Correlation (QUIC)

Material state awareness of composites using conventional Nondestructive Evaluation (NDE) method is limited by finding the size and the locations of the cracks and the delamination in a composite structure. To aid the progressive failure models using the slow growth criteria, the awareness of the precursor damage state and quantification of the degraded material properties is necessary, which is challenging using the current NDE methods. To quantify the material state, a new offline NDE method is reported herein. The new method named Quantitative Ultrasonic Image Correlation (QUIC) is devised, where the concept of microcontinuum mechanics is hybrid with the experimentally measured Ultrasonic wave parameters. This unique combination resulted in a parameter called Nonlocal Damage Entropy for the precursor awareness. High frequency (more than 25 MHz) scanning acoustic microscopy is employed for the proposed QUIC. Eight woven carbon-fiber-reinforced-plastic composite specimens were tested under fatigue up to 70% of their remaining useful life. During the first 30% of the life, the proposed nonlocal damage entropy is plotted to demonstrate the degradation of the material properties via awareness of the precursor damage state. Visual proofs for the precursor damage states are provided with the digital images obtained from the micro-optical microscopy, the scanning acoustic microscopy and the scanning electron microscopy

Numerical Methods and Algorithms for High Frequency Wave Scattering Problems in Homogeneous and Random Media

This dissertation consists of four integral parts with a unified objective of developing efficient numerical methods for high frequency time-harmonic wave equations defined on both homogeneous and random media. The first part investigates the generalized weak coercivity of the acoustic Helmholtz, elastic Helmholtz, and time-harmonic Maxwell wave operators. We prove that such a weak coercivity holds for these wave operators on a class of more general domains called generalized star-shape domains. As a by-product, solution estimates for the corresponding Helmholtz-type problems are obtained. The second part of the dissertation develops an absolutely stable (i.e. stable in all mesh regimes) interior penalty discontinuous Galerkin (IP-DG) method for the elastic Helmholtz equations. A special mesh-dependent sesquilinear form is proposed and is shown to be weakly coercive in all mesh regimes. We prove that the proposed IP-DG method converges with optimal rate with respect to the mesh size. Numerical experiments are carried out to demonstrate the theoretical results and compare this method to the standard finite element method. The third part of the dissertation develops a Monte Carlo interior penalty discontinuous Galerkin (MCIP-DG) method for the acoustic Helmholtz equation defined on weakly random media. We prove that the solution to the random Helmholtz problem has a multi-modes expansion (i.e., a power series in a medium- related small parameter). Using this multi-modes expansion an efficient and accurate numerical method for computing moments of the solution to the random Helmholtz problem is proposed. The proposed method is also shown to converge optimally. Numerical experiments are carried out to compare the new multi-modes MCIP-DG method to a classical Monte Carlo method. The last part of the dissertation develops a theoretical framework for Schwarz pre- conditioning methods for general nonsymmetric and indefinite variational problems which are discretized by Galerkin-type discretization methods. Such a framework has been missing in the literature and is of great theoretical and practical importance for solving convection-diffusion equations and Helmholtz-type wave equations. Condition number estimates for the additive and hybrid Schwarz preconditioners are established under some structure assumptions. Numerical experiments are carried out to test the new framework

Hyperholomorphic structures and corresponding explicit orthogonal function systems in 3D and 4D

Die Reichhaltigkeit und breite Anwendbarkeit der Theorie der holomorphen Funktionen in der komplexen Ebene ist stark motivierend eine ähnliche Theorie für höhere Dimensionen zu entwickeln. Viele Forscher waren und sind in diese Aufgaben involviert, insbesondere in der Entwicklung der Quaternionenanalysis. In den letzten Jahren wurde die Quaternionenanalysis bereits erfolgreich auf eine Vielzahl von Problemen der mathematischen Physik angewandt. Das Ziel der Dissertation besteht darin, holomorphe Strukturen in höheren Dimensionen zu studieren. Zunächst wird ein neues Holomorphiekonzept vorgelegt, was auf der Theorie rechtsinvertierbarer Operatoren basiert und nicht auf Verallgemeinerungen des Cauchy-Riemann-Systems wie üblich. Dieser Begriff umfasst die meisten der gut bekannten holomorphen Strukturen in höheren Dimensionen. Unter anderem sind die üblichen Modelle für reelle und komplexe quaternionenwertige Funktionen sowie Clifford-algebra-wertige Funktionen enthalten. Außerdem werden holomorphe Funktionen mittels einer geeignete Formel vom Taylor-Typ durch spezielle Funktionen lokal approximiert. Um globale Approximationen für holomorphe Funktionen zu erhalten, werden im zweiten Teil der Arbeit verschiedene Systeme holomorpher Basisfunktionen in drei und vier Dimensionen mittels geeigneter Fourier-Entwicklungen explizit konstruiert. Das Konzept der Holomorphie ist verbunden mit der Lösung verallgemeinerter Cauchy-Riemann Systeme, deren Funktionswerte reellen Quaternionen bzw. reduzierte Quaternionen sind. In expliziter Form werden orthogonale holomorphe Funktionensysteme konstruiert, die Lösungen des Riesz-Systems bzw. des Moisil-Teodorescu Systems über zylindrischen Gebieten im R3, sowie Lösungen des Riesz-Systems in Kugeln des R4 sind. Um konkrete Anwendungen auf Randwertprobleme realisieren zu können wird eine orthogonale Zerlegung eines Rechts-Quasi-Hilbert-Moduls komplex-quaternionischer Funktionen unter gegebenen Bedingungen studiert. Die Ergebnisse werden auf die Behandlung von Maxwell-Gleichungen mit zeitvariabler elektrischer Dielektrizitätskonstante und magnetischer Permeabilität angewandt.The richness and widely applicability of the theory of holomorphic functions in complex analysis requires to perform a similar theory in higher dimensions. It has been developed by many researchers so far, especially in quaternionic analysis. Over the last years, it has been successfully applied to a vast array of problems in mathematical physics. The aim of this thesis is to study the structure of holomorphy in higher dimensions. First, a new concept of holomorphy is introduced based on the theory of right invertible operators, and not by means of an analogue of the Cauchy-Riemann operator as usual. This notion covers most of the well-known holomorphic structures in higher dimensions including real, complex, quaternionic, Clifford analysis, among others. In addition, from our operators a local approximation of a holomorphic function is attained by the Taylor type formula. In order to obtain the global approximation for holomorphic functions, the second part of the thesis deals with the construction of different systems of basis holomorphic functions in three and four dimensions by means of Fourier analysis. The concept of holomorphy is related to the null-solutions of generalized Cauchy-Riemann systems, which take either values in the reduced quaternions or real quaternions. We obtain several explicit orthogonal holomorphic function systems: solutions to the Riesz and Moisil-Teodorescu systems over cylindrical domains in R3, and solutions to the Riesz system over spherical domains in R4. Having in mind concrete applications to boundary value problems, we investigate an orthogonal decomposition of complex-quaternionic functions over a right quasi-Hilbert module under given conditions. It is then applied to the treatment of Maxwell’s equations with electric permittivity and magnetic permeability depending on the time variable