Analytical properties of solutions of the Schrödinger equation and quantization of charge

The Schwinger--DeWitt expansion for the evolution operator kernel is used to investigate analytical properties of the Schr\"odinger equation solution in time variable. It is shown, that this expansion, which is in general asymptotic, converges for a number of potentials (widely used, in particular, in one-dimensional many-body problems), and besides, the convergence takes place only for definite discrete values of the coupling constant. For other values of charge the divergent expansion determines the functions having essential singularity at origin (beyond usual \delta-function). This does not permit one to fulfil the initial condition. So, the function obtained from the Schr\"odinger equation cannot be the evolution operator kernel. The latter, rigorously speaking, does not exist in this case. Thus, the kernel exists only for definite potentials, and moreover, at the considered examples the charge may have only quantized values

Experimental and numerical analysis of the elastic behaviour of the TRIP 700 steel for springback predictions

The demand for lightweight fuel-efficient vehicles and the increasing industrial competitiveness between constructors have motivated the automotive industry to introduce new materials and to reinforce the accuracy of metal forming processes with the aim of reducing costs. Steady advancements in technology, combined with shortened product cycle times and continual demand to reduce costs, have resulted in the dependence on finite element (FE) codes for the simulation of sheet metal forming processes in the automotive industry. It is generally accepted that commercially available FE programs can be successfully utilised to predict formability and the likelihood of splits and wrinkles, even for complex part geometry. However, the prediction of the elastically-driven change of shape after forming a product (springback) remains a challenge. This inaccuracy of springback prediction involves a need to use expensive and time consuming experimental try-outs to determine the proper tool geometry and all other parameters which can lead to the desired final shape. The main objective of the research presented in this thesis has been to improve the current numerical algorithms and material modelling techniques that allow the industry to reach a more accurate springback predictions in sheet metal forming. The literature review shows the relevance of the elastic behaviour on springback predictions. Therefore, first, a material characterisation of the elastic and elastoplastic behaviours has been performed. Aiming at accurately characterising the material, tensile tests, loading-unloading cyclic tests and tension-compression tests have been carried out. In order to improve the elastic behaviour model's accuracy, the classical elastic law has been extended. By means of this extension the elastic model is able to represent both linear and non-linear behaviours. Therefore, it has been possible to introduce the elastic behaviour with different levels of accuracy to predict the springback. Aiming at analysing the improvement introduced by modelling more accurately the elastic behaviour of the sheet, different simulations have been compared with experimental data. V-free bending experimental tests have been conducted in an AHSS sheet steel. Each numerical simulations has been provided with different particularisation of the extended model, representing each one the elastic behaviour of the material with different levels of accuracy. The results show the importance of an accurate modelling of the elastic behaviour for springback prediction.Ingurumen legediak errespetatzeko ibilgailuen pisua gutxitze beharra eta merkatuaren lehiakortasuna dela eta, automobil industriak material berriak erabiltzera eta produkzio prozesuak hobetzera behartua izan da. Gaur egun, elementu nitu bidezko kalkuluak ezinbestekoak bihurtu dira automobil industrian prozesuen konplexutasuna ikertzeko eta hauek optimizatzeko, horrela kostuak gutxitzen direlarik. Elementu nituetako softwareen zehaztasuna industriagatik berretsia izan da. Kode hauek xehetasunez iragarri dezakete landutako materialaren forma eta honetan agertuko diren akatsak prozesuaren kon gurazioaren arabera. Hala eta guztiz ere, materiala landu ostean gertatzen den berreskuratze elastikoa (springback) aurresatea oraindik lortugabeko erronka bat da. Zehaztasun falta honek garestia den test-akats teknika bat burutzera behartzen du industria, behar den forma lortzeko fabrikazio prozesua egokitzen delarik. Ikerketa lan honen helburu nagusia springbacka iragartzeko erabiltzen diren material modeloak eta teknika matematikoak hobetzea da. Hobetze honen bitartez, gaur egungo prozesuen optimizazioa hobetzea espero da, iragarpenen zehaztasuna areagotuz, modu honetan kostuak murrizteko. Burututako bibliogra a ikerketaren bitartez, materialaren portaera elastikoaren garrantzia springbacka aurreikusteko ezarri egin izan da. Karakterizazio zehatz bat lortzeko trakzio, trakzio-erlaxazio eta trakzio-konpresio saiakuntzak burutu dira. Portaera elastikoaren zehaztasunaren eragina aztertzeko helburuarekin lege elastiko klasikoaren luzapen bat proposatu da. Luzapen honen bitartez, modelo elastiko berria, portaera lineala zein ez linealak irudikatzeko gai da. Luzapen hau erabiliz zehaztasun desberdineko portaera elastikoak sartu ahal izan dira springback iragarpen simulazioetan. Modelo elastiko zehatz batek springback iragarpenean egindako hobekuntza zenbakiko modelo desberdinak saiakuntza sartu esperimentalekin alderatuz aztertua izan da. Modelo bakoitzak zehaztasun desberdinekin adierazi du portaera elastikoa, lege elastikoaren luzapena erabiliz. Iragarpen hauek alderatu dira erresistentzia handiko altzairu baten burututako V-bending saiakuntza baten datu esperimentalekin. Lan honetatik ondorioztatzen da springback iragarpenentzako portaera elastikoaren irudikapen zehatzaren garrantzia.La necesidad de aligerar los vehículos para reducir el consumo y el aumento de la competitividad ha motivado a la industria del automóvil a introducir nuevas familias de materiales y reforzar la precisión de los procesos de fabricación con el objetivo de reducir gastos. La complejidad de los procesos de conformado, junto con la demanda de disminución de costes, ha conllevado a la dependencia de códigos de elementos finitos para la simulación y optimización de los procesos en la industria de la automoción. La precisión de los software comerciales de elementos finitos ha sido ampliamente validada industrialmente. Estos son capaces de predecir con exactitud la conformabilidad, defectos y secciones críticas incluso para piezas complejas. Sin embargo, la predicción de la recuperación elástica que se produce tras el conformado(también llamado springback) aún es un desafío sin resolver. La inexactitud en la predicción de springbcak conlleva la necesidad de un costoso método de prueba-error para la puesta a punto de los procesos y de los utillajes. El principal objetivo de esta tesis es la mejora de los algoritmos numéricos y de los modelos de material para la predicción de springback a nivel industrial en el conformado de chapa. De la revisión bibliográfica se desprende la importancia del comportamiento elástico en la predicción del springback. Con la intención de realizar una caracterización exhaustiva se han llevado a cabo ensayos de tracción, ensayos de ciclos de tracción-relajación y ensayos de tensión-compresión. A fin de analizar la influencia de la precisión del modelo de comportamiento elástico se propone una extensión de la ley elástica clásica. Mediante esta extensión el modelo elástico es capaz de representar tanto comportamientos lineales como no-lineales. Haciendo uso de este nuevo modelo, ha sido posible introducir diferentes niveles de precisión del comportamiento elástico en las simulaciones de springback. La mejora introducida en la predicción del springbak debida a la precisión del modelo elástico ha sido analizada comparando diferentes modelos numéricos con datos experimentales. Cada modelo ha representado el comportamiento elástico con diferentes precisiones haciendo uso de la extensión de la ley elástica. Estas predicciones han sido comparadas con los datos experimentales de un ensayo de V-bending realizado empleando un acero de alta resistencia. De este trabajo se concluye la importancia de una representación precisa del comportamiento elástico para predicciones de springback

Solution strategies for nonlinear conservation laws

Nonlinear conservation laws form the basis for models for a wide range of physical phenomena. Finding an optimal strategy for solving these problems can be challenging, and a good strategy for one problem may fail spectacularly for others. As different problems have different challenging features, exploiting knowledge about the problem structure is a key factor in achieving an efficient solution strategy. Most strategies found in literature for solving nonlinear problems involve a linearization step, usually using Newton's method, which replaces the original nonlinear problem by an iteration process consisting of a series of linear problems. A large effort is then spent on finding a good strategy for solving these linear problems. This involves choosing suitable preconditioners and linear solvers. This approach is in many cases a good choice and a multitude of different methods have been developed. However, the linearization step to some degree involves a loss of information about the original problem. This is not necessarily critical, but in many cases the structure of the nonlinear problem can be exploited to a larger extent than what is possible when working solely on the linearized problem. This may involve knowledge about dominating physical processes and specifically on whether a process is near equilibrium. By using nonlinear preconditioning techniques developed in recent years, certain attractive features such as automatic localization of computations to parts of the problem domain with the highest degree of nonlinearities arise. In the present work, these methods are further refined to obtain a framework for nonlinear preconditioning that also takes into account equilibrium information. This framework is developed mainly in the context of porous media, but in a general manner, allowing for application to a wide range of problems. A scalability study shows that the method is scalable for challenging two-phase flow problems. It is also demonstrated for nonlinear elasticity problems. Some models arising from nonlinear conservation laws are best solved using completely different strategies than the approach outlined above. One such example can be found in the field of surface gravity waves. For special types of nonlinear waves, such as solitary waves and undular bores, the well-known Korteweg-de Vries (KdV) equation has been shown to be a suitable model. This equation has many interesting properties not typical of nonlinear equations which may be exploited in the solver, and strategies usually reserved to linear problems may be applied. In this work includes a comparative study of two discretization methods with highly different properties for this equation

Minimal Nodal Domains for Strictly Elliptic Partial Differential Equations with Homogeneous Boundary Conditions

This work presents a proof of the dependence of the first eigenvalue for uniformly elliptic partial differential equations on the domain in a less abstract setting than that of Ivo Babushka and Rudolf Vyborny in 1965. The proof contained here, under rather mild conditions on the boundary of the domain, �Ω, demonstrates that the first eigenvalue of elliptic partial differential equation [ �� + �� = 0 �� Ω [ � = 0 �� �Ω depends continuously on the domain in the following sense. If a sequence of domains is such that, then the corresponding first eigenvalues satisfy is the first eigenvalue for [ �� + �� = 0 �� Ω [ � = 0 �� �Ω The work also reviews and utilizes the Sturmian comparison results of John G. Heywood, E. S. Noussair, and Charles A. Swanson. For a continuously parameterized family of domains, say with μ ∈ = [a, b], the continuous dependence of the eigenvalue on the domain combined with the Sturmian comparison results provide a theorem that insures, under certain conditions, that the elliptic partial differential equation [ �� = 0 �� Ω [ � = 0 �� �Ω has a solution which is positive on a nodal domain That is there is a least value of μ [a, b] so that a positive solution u exists for [ �� = 0 �� Ωμ [ � = 0 �� �Ωμ Beyond these results the work contains a theorem that shows for certain types of domains, rectangles in , among them, that there is a critical dimension smaller than which, no solution to the problem [ �� + �� = 0 �� Ω [ � = 0 �� �Ω exists when the eigenvalue is fixed. During the investigations taken up in this work, certain observations were made regarding linear approximations to eigenvalue problems in R2 using a standard numerical approximation scheme. One such observation is that if a linear approximation to an eigenvalue problem contains an incorrect estimate for an eigenvalue, the resulting graphical approximation seems to betray whether or not the estimate was low or high. The observations made do not appear to exist in the literature

Piecewise Linear Wavelet Collocation on Triangular Grids, Approximation of the Boundary Manifold and Quadrature

In this paper we consider a piecewise linear collocation method for the solution of a pseudo-differential equations of order r = 0,-1 over a closed and smooth boundary manifold. The trial space is the space of all continuous and piecewise linear functions defined over a uniform triangular grid and the collocation points are the grid points. For the wavelet basis in the trial space we choose the three-point hierarchical basis together with a slight modification near the boundary points of the global patches of parametrization. We choose three, four, and six term linear combinations of Dirac delta functionals as wavelet basis in the space of test functionals. Though not all wavelets have vanishing moments, we derive the usual compression results, i.e. we prove that, for N degrees of freedom, the fully populated stiffness matrix of N2 entries can be approximated by a sparse matrix with no more than O(N [log N]2.25) non-zero entries. The main topic of the present paper, however, is to show that the parametrization can be approximated by low order piecewise polynomial interpolation and that the integrals in the stiffness matrix can be computed by quadrature, where the quadrature rules are combinations of product integration applied to non analytic factors of the integrand and of high order Gau{\ss} rules applied to the analytic parts. The whole algorithm for the assembling of the matrix requires no more than O(N [log N]4.25) arithmetic operations, and the error of the collocation approximation, including the compression, the approximative parametrization, and the quadratures, is less than O(N-1[log N]2). Note that, in contrast to well-known algorithms by v.Petersdorff, Schwab, and Schneider, only a finite degree of smoothness is required

Fractional evolution equations in Banach spaces

IV+107hlm.;24c

The calculus according to S. F. Lacroix (1765-1843)

Silvestre François Lacroix (Paris. 1765 - ibid., 1843) was not a prominent mathematical researcher, but he was certainly a most influential mathematical book author. His most famous book is a monumental Traité du calcul différentiel et du calcul intégral (three large volumes, 1797-1800; a second edition appeared in 1810-1819) - an encyclopaedic appraisal of 18th-century calculus. He also published many textbooks, one of which is closely associated to this large Traité: the Traité élémentaire du calcul différentiel et du calcul intégral (first edition in 1802; four more editions in Lacroix's lifetime; four posthumous editions). Although most historians acknowledge the great influence of Lacroix's large Traité in early 19th-century mathematics it has not been thoroughly studied. This thesis is a contribution for correcting this omission. The focus is on its first edition, but the second edition and the Traité élémentaire, are also addressed. The thesis starts with a short biography of Lacroix, followed by an overview of the first edition of the large Traité. Next corne five chapters where particular aspects are analyzed in detail: the foundations of the calculus, analytic and differential geometry, approximate integration and conceptions of the integral, types of solutions of differential equations (singular/complete/general integrals, geometrical interpretations, and generality of arbitrary functions), and three aspects related to finite differences and series (the use of subscript indices, types of solutions of finite difference equations, and mixed difference equations); for all these aspects Lacroix's treatment is compared to the 18th-century background, and to his likely sources. Then we examine how the large Traité was adapted to a textbook - the Traité élémentaire, we take a look at the second edition of the large Traité, and conclude the body of the thesis with some final remarks