La méthode d’imagerie la plus répandue dans l’industrie pétrolière est la RTM (Reverse Time Migration) qui repose sur la simulation de la propagation des ondes dans le sous-sol. Nous nous sommes concentrés sur un propagateur d'ondes élastiques 3D en milieu anisotrope de type TTI (Tilted Transverse Isotropic). Nous avons directement travaillé dans le code de recherche de Total DIVA (Depth Imaging Velocity Analysis), basé sur une discrétisation par la méthode de Galerkin Discontinue et le schéma Leap-Frog, et développé pour le calcul parallèle intensif – HPC (High Performance Computing). Nous avons ciblé plus particulièrement deux contributions possibles qui, si elles supposent des compétences très différentes, ont la même finalité : réduire les coûts de calculs requis pour la simulation. D'une part, les conditions aux limites classiques de type PML (Perfectly Matched Layers) ne sont pas stables dans des milieux TTI. Nous avons proposé de formuler une CLA (Conditions aux Limites Absorbantes) stable dans des milieux anisotropes. La méthode de construction repose sur les propriétés des courbes de lenteur, ce qui donne à notre approche un caractère original. D'autre part, le parallélisme initial, basé sur une décomposition de domaine et des communications par passage de messages à l'aide de la bibliothèque MPI, conduit à un déséquilibrage de charge qui détériore son efficacité parallèle. Nous avons corrigé cela en remplaçant le paradigme parallélisme par l'utilisation de la programmation à base de tâches sur support d'exécution. Cette thèse a été réalisée dans le cadre de l'action de recherche DIP (Depth Imaging Partnership) qui lie la compagnie pétrolière Total et Inria. / The most common method of Seismic Imaging is the RTM (Reverse Time Migration) which depends on wave propagation simulations in the subsurface. We focused on a 3D elastic wave propagator in anisotropic media, more precisely TTI (Tilted Transverse Isotropic). We directly worked in the Total code DIVA (Depth Imaging Velocity Analysis) which is based on a discretization by the Discontinuous Galerkin method and the Leap-Frog scheme, and developed for intensive parallel computing – HPC (High Performance Computing). We choose to especially target two contributions. Although they required very different skills, they share the same goal: to reduce the computational cost of the simulation. On one hand, classical boundary conditions like PML (Perfectly Matched Layers) are unstable in TTI media. We have proposed a formulation of a stable ABC (Absorbing Boundary Condition) in anisotropic media. The technique is based on slowness curve properties, giving to our approach an original side. On the other hand, the initial parallelism, which is based on a domain decomposition and communications by message passing through the MPI library, leads to load-imbalance and so poor parallel efficiency. We have fixed this issue by replacing the paradigm for parallelism by the use of task-based programming through runtime system. This PhD thesis have been done in the framework of the research action DIP (Depth Imaging Partnership) between the Total oil company and Inria.

This study focused on experimentally determining the cooling performance of a fully cooled, scaled-up model of a C3X turbine vane. The primary objective was to determine the differences in overall effectiveness in the presence and absence of a hot mainstream flowing over the vane. Overall effectiveness was measured using a thermally scaled matched Biot number vane with an impingement plate providing the internal cooling. This is the first study focused on investigating the effect of removing the mainstream flow and comparing the contour and laterally-averaged effectiveness data in support of the development of an assembly line thermal testing method. It was found that the proposed method of factory floor testing of turbine component cooling performance did not provide comparable information to traditional overall effectiveness test methods. A second experiment was performed in which the effect of altering the angle of attack of a flow into a passive turbulence generator was investigated. Measurements in the approach flow were taken using a single wire hot-wire anemometer. This study was the first to investigate the effects such a setup would have on fluctuating flow quantitates such as turbulence intensity and integral length scale rather than simply the mean quantities. It was found that both the downstream turbulence intensity and the turbulence integral length scale increase monotonically with approach flow incidence angle at a specified distance downstream of the turbulence generator. / text

Darbe nagrinėjamos simetrinės trečiosios eilės liestinės sluoksniuotės, kurios apibrėžiamos kaip 3 - džetų aibės. Surasta simetrinės erdvės izotropijų grupė, o taip pat jos izomorfijų grupė. Gautos izomorfijų grupės struktūrinės lygtys, surasti erdvės Maurerio – Kartano lygčių analogai, įrodytos formulės, išreiškiančios indukuotosios afiniosios sieties kreivumo tenzorių komponentes izomorfijų grupės struktūrinėmis konstantomis. Taip pat gauta visa eilė tapatybių, siejančių kreivumo objektus ir izomorfijų grupės struktūrines konstantas (apibendrintos Ričio ir Bianchi tapatybės). / The paper examined the symmetric third order tangent bundle, defined as 3- jet space. Found symmetric space isotropy group, as well as its isomorphy group. The resulting structural equation of isomorphy group, find this area Maurer - Cartan analogues of equations, an established formula, expressing inducted affines connection component of curvature tensors of the isomorphy group structural constants. Also received identities connecting the curvature objects structural constants of isomorphy group (generalized in Riči and Bianchi identity).

The topic of this thesis is the numerical simulation of transversely isotropic 3D structures and laminates by means of the adaptive finite element method. To achieve this goal, the theoretical background of elastic deformation problems, transverse isotropy, plate theory, and the classical laminate theory is recapitulated. The classical laminate theory implies a combination of the membrane problem and the plate problem with additional coupling terms. The focus of this work is the adjustment of two integral parts of the adaptive FE algorithm according to the classical laminate theory. One of these parts is the solution of the FE system; a good preconditioner is needed in order to use the conjugate gradient method efficiently. It is shown via a spectral equivalence bound that the combination of existing preconditioners for the membrane and plate problems poses a capable preconditioner for the combined laminate problem. The other part is the error estimation process; the error estimator determines where the current mesh has to be refined for the next step. Existing results on residual error estimators for the elasticity problem, the biharmonic problem, and the plate problem are combined and extended to obtain a posteriori local residual error indicators for the classical laminate theory problem. The effectiveness of both results is demonstrated by numerical examples.

Les données provenant de l'échantillonnage fin d'un processus continu (champ aléatoire) peuvent être représentées sous forme d'images. Un test statistique permettant de détecter une différence entre deux images peut être vu comme un ensemble de tests où chaque pixel est comparé au pixel correspondant de l'autre image. On utilise alors une méthode de contrôle de l'erreur de type I au niveau de l'ensemble de tests, comme la correction de Bonferroni ou le contrôle du taux de faux-positifs (FDR). Des méthodes d'analyse de données ont été développées en imagerie médicale, principalement par Keith Worsley, utilisant la géométrie des champs aléatoires afin de construire un test statistique global sur une image entière. Il s'agit d'utiliser l'espérance de la caractéristique d'Euler de l'ensemble d'excursion du champ aléatoire sous-jacent à l'échantillon au-delà d'un seuil donné, pour déterminer la probabilité que le champ aléatoire dépasse ce même seuil sous l'hypothèse nulle (inférence topologique). Nous exposons quelques notions portant sur les champs aléatoires, en particulier l'isotropie (la fonction de covariance entre deux points du champ dépend seulement de la distance qui les sépare). Nous discutons de deux méthodes pour l'analyse des champs anisotropes. La première consiste à déformer le champ puis à utiliser les volumes intrinsèques et les compacités de la caractéristique d'Euler. La seconde utilise plutôt les courbures de Lipschitz-Killing. Nous faisons ensuite une étude de niveau et de puissance de l'inférence topologique en comparaison avec la correction de Bonferroni. Finalement, nous utilisons l'inférence topologique pour décrire l'évolution du changement climatique sur le territoire du Québec entre 1991 et 2100, en utilisant des données de température simulées et publiées par l'Équipe Simulations climatiques d'Ouranos selon le modèle régional canadien du climat. / Data coming from a fine sampling of a continuous process (random field) can be represented as images. A statistical test aiming at detecting a difference between two images can be seen as a group of tests in which each pixel is compared to the corresponding pixel in the other image. We then use a method to control the type I error over all the tests, such as the Bonferroni correction or the control of the false discovery rate (FDR). Methods of data analysis have been developped in the field of medical imaging, mainly by Keith Worsley, using the geometry of random fields in order to build a global statistical test over the whole image. The expected Euler characteristic of the excursion set of the random field underlying the sample over a given threshold is used in order to determine the probability that the random field exceeds this same threshold under the null hypothesis (topological inference). We present some notions relevant to random fields, in particular isotropy (the covariance function between two given points of a field depends only on the distance between them). We discuss two methods for the analysis of non\-isotropic random fields. The first one consists in deforming the field and then using the intrinsic volumes and the Euler characteristic densities. The second one uses the Lipschitz-Killing curvatures. We then perform a study of sensitivity and power of the topological inference technique comparing it to the Bonferonni correction. Finally, we use topological inference in order to describe the evolution of climate change over Quebec territory between 1991 and 2100 using temperature data simulated and published by the Climate Simulation Team at Ouranos, with the Canadian Regional Climate Model CRCM4.2.

Μια πολλαπλότητα Riemann (M, g) ονομάζεται Einstein αν έχει σταθερή καμπυλότητα Ricci. Είναι γνωστό ότι αν (M=G/K, g) είναι μια συμπαγής ομογενής πολλαπλότητα Riemann, τότε οι G-αναλλοίωτες μετρικές Einstein μοναδιαίου όγκου, είναι τα κρίσιμα σημεία του συναρτησοειδούς ολικής βαθμωτής καμπυλότητας περιορισμένο στο χώρο των G-αναλλοίωτων μετρικών με όγκο 1. Για μια G-αναλλοίωτη μετρική Riemann η εξίσωση Einstein ανάγεται σε ένα σύστημα αλγεβρικών εξισώσεων. Οι θετικές πραγματικές λύσεις του συστήματος αυτού είναι ακριβώς οι G-αναλλοίωτες μετρικές Einstein που δέχεται η πολλαπλότητα Μ. Μια σημαντική οικογένεια συμπαγών ομογενών χώρων αποτελείται από τις γενικευμένες πολλαπλότητες σημαιών. Κάθε τέτοιος χώρος είναι μια τροχιά της συζυγούς αναπαράστασης μιας συμπαγούς, συνεκτικής, ημι-απλής ομάδας Lie G. Πρόκειται για ομογενείς πολλαπλότητες της μορφής G/C(S), όπου C(S) είναι ο κεντροποιητής ενός δακτυλίου S στην G. Κάθε τέτοιος χώρος δέχεται ένα πεπερασμένο πλήθος από G-αναλλοίωτες μετρικές Kahler-EInstein. Στην παρούσα διατριβή ταξινομούμε όλες τις πολλαπλότητες σημαιών G/K που αντιστοιχούν σε μια απλή ομάδα Lie G, των οποίων η ισοτροπική αναπαράσταση διασπάται σε 2 ή 4 μη αναγώγιμους και μη ισοδύναμους Ad(K)-αναλλοίωτους προσθετέους. Για κάθε τέτοιο χώρο λύνουμε την αναλλοίωτη εξίσωση Εinstein, και παρουσιάζουμε την αναλυτική μορφή νέων G-αναλλοίωτων μετρικών Einstein. Στις περισσότερες περιπτώσεις παρουσιάζουμε την πλήρη ταξινόμηση των αναλλοίωτων μετρικών Einstein. Επίσης εξετάζουμε το ισομετρικό πρόβλημα. Για την κατασκευή της εξίσωσης Einstein σε κάποιες πολλαπλότητες σημαιών με 4 ισοτροπικούς προσθετέους χρησιμοποιούμε την νηματοποίηση συστροφής που δέχεται κάθε πολλαπλότητα σημαιών επί ενός ισοτροπικά μη αναγώγιμου συμμετρικού χώρου συμπαγούς τύπου. Αυτή η μέθοδος είναι καινούργια και μπορεί να εφαρμοστεί και σε άλλες πολλαπλότητες σημαιών. / A Riemannian manifold (M, g) is called Einstein, if it has constant Ricci curvature. It is well known that if (M=G/K, g) is a compact homogeneous Riemannian manifold, then the G-invariant \tl{Einstein} metrics of unit volume, are the critical points of the scalar curvature function restricted to the space of all G-invariant metrics with volume 1. For a G-invariant Riemannian metric the Einstein equation reduces to a system of algebraic equations. The positive real solutions of this system are the $G$-invariant Einstein metrics on M. An important family of compact homogeneous spaces consists of the generalized flag manifolds. These are adjoint orbits of a compact semisimple Lie group. Flag manifolds of a compact connected semisimple Lie group exhaust all compact and simply connected homogeneous Kahler manifolds and are of the form G/C(S), where C(S) is the centralizer (in G) of a torus S in G. Such homogeneous spaces admit a finite number of G-invariant complex structures, and for any such complex structure there is a unique compatible G-invariant Kahler-Einstein metric. In this thesis we classify all flag manifolds M=G/K of a compact simple Lie group G, whose isotropy representation decomposes into 2 or 4, isotropy summands. For these spaces we solve the (homogeneous) Einstein equation, and we obtain the explicit form of new G-invariant Einstein metrics. For most cases we give the classification of homogeneous Einstein metrics. We also examine the isometric problem. For the construction of the Einstein equation on certain flag manifolds with four isotropy summands, we apply for first time the twistor fibration of a flag manifold over an isotropy irreducible symmetric space of compact type. This method is new and it can be used also for other flag manifolds. For flag manifolds with two isotropy summands, we use the restricted Hessian and we characterize the new Einstein metrics as local minimum points of the scalar curvature function restricted to the space of G-invariant Riemannian metrics of volume 1. We mention that the classification of flag manifolds with two isotropy summands gives us new examples of homogeneous spaces, for which the motion of a charged particle under the electromagnetic field, and the geodesics curves, are completely determined.

Made available in DSpace on 2014-06-11T19:27:13Z (GMT). No. of bitstreams: 0 Previous issue date: 2008-09-03Bitstream added on 2014-06-13T18:31:04Z : No. of bitstreams: 1 yamasaki_mi_me_ilha.pdf: 16778400 bytes, checksum: 6afbdce5c51d7a7040bbcdba50a7949f (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / É apresentado um estudo experimental da laminação de tiras fundidas a partir do material semi-sólido obtido na calha de resfriamento que alimenta continuamente um laminador duo. Os cilindros do laminador estão posicionados horizontalmente e podem ser operados na velocidade de 0,25 m/s, 0,47 m/s, 0,73 m/s e 1,07 m/s. A velocidade de 0,25 m/s produziu uma tira de melhor qualidade. Ligas hipoeutéticas Pb/Sn (Pb–30%Sn, Pb-40%Sn, Pb-50%Sn) e próxima ao ponto eutético (Pb-63%Sn), respectivamente, com intervalo de solidificação de 75 °C, 56 °C, 31 °C e 6 °C de acordo com o diagrama de fases, foram utilizadas nos ensaios experimentais para obter tiras semi-sólidas fundidas e tixoconformadas para comparação. As diversas simulações usando as ligas de Pb/Sn têm revelado a importância do intervalo de solidificação e temperatura de vazamento da liga, da velocidade dos cilindros, da temperatura do bocal junto ao cilindro inferior, da superfície de acabamento dos cilindros e da geometria da panela intermediária (tundish), sobre a qualidade do produto final. A liga Pb-30%Sn com alto intervalo de solidificação em comparação com outras ligas testadas, apresentou maior dificuldade para ser tixolaminada. Isso ocorreu, porque as ligas de alto intervalo de solidificação tendem a formar trincas à quente no final da solidificação. Como resultado, uma pasta metálica plástica é difícil de formar. O caminho provável para obter uma tira semi-sólida fundida de boa qualidade neste caso, é aplicar uma inoculação que produz grãos finos antes do vazamento. O controle para a tixolaminação empregando a liga Sn-37%Pb com intervalo de solidificação menor, e elevada fluidez, é mais rigoroso para obter uma tira contínua. Conseqüentemente, foram utilizadas diferentes temperaturas de vazamento (260, 240 e 220 ºC) para controlar a fluidez e obter o tempo de contato... / This is an experimental study of cast strip rolling from semi-solid material employing a cooling slope which continuously feeds a rolling mill. The cylinders of the rolling mill are positioned horizontally and can be operated at speeds of 0.25 m/s, 0.47 m/s, 0.73 m/s and 1.07 m/s. The lower speed of 0,25 m/s produces a strip of better quality. Hypoeutectic Pb/Sn alloys (Pb-30%Sn, Pb-40%Sn, Pb-50%Sn) and near eutectic point alloys (Pb-63%Sn), with solidification intervals of 75°C, 56°C, 31°C and 6°C respectively, according to the phase diagram, were used in experimental tests to obtain cast semi-solid and thixorolled strips for comparison. Simulations highlighted the necessary control parameters required to obtain good quality of the strip. These were: control alloy solidification interval, pouring temperature, roll speeds, ceramic nozzle temperature at the lower roll, quality of the roll surface finishing and tundish geometry. The Pb-30%Sn alloy, which has a much higher solidification interval in comparison with the other alloys tested, was difficult to thixoroll. This is because alloys with a high solidification interval tend to form hot tears at the end of solidification, and prevent a plastic metallic mush from forming. The probable solution to obtaining a semi-solid fused strip of good quality with this material, is to apply an inoculation that produces fine grains just before the pouring. In contrast, the parameter control for thixorolling of the Sn-37%Pb alloy, with lesser solidification interval and elevated fluidity, needed to be rigorous to obtain a continuous strip. Consequently, several pouring temperatures (260, 240 and 220ºC) were used to vary the fluidity and obtain sufficient alloy-inferior cylinder contact time for complete solidification. The strips obtained by the twin and single roll processing, and conventional rolling were characterized... (Complete abstract click electronic access below)

Orientador: Antônio de Pádua Lima Filho / Banca: João Batista Campos Silva / Banca: Alcides Padilha / Resumo: É apresentado um estudo experimental da laminação de tiras fundidas a partir do material semi-sólido obtido na calha de resfriamento que alimenta continuamente um laminador duo. Os cilindros do laminador estão posicionados horizontalmente e podem ser operados na velocidade de 0,25 m/s, 0,47 m/s, 0,73 m/s e 1,07 m/s. A velocidade de 0,25 m/s produziu uma tira de melhor qualidade. Ligas hipoeutéticas Pb/Sn (Pb-30%Sn, Pb-40%Sn, Pb-50%Sn) e próxima ao ponto eutético (Pb-63%Sn), respectivamente, com intervalo de solidificação de 75 °C, 56 °C, 31 °C e 6 °C de acordo com o diagrama de fases, foram utilizadas nos ensaios experimentais para obter tiras semi-sólidas fundidas e tixoconformadas para comparação. As diversas simulações usando as ligas de Pb/Sn têm revelado a importância do intervalo de solidificação e temperatura de vazamento da liga, da velocidade dos cilindros, da temperatura do bocal junto ao cilindro inferior, da superfície de acabamento dos cilindros e da geometria da panela intermediária (tundish), sobre a qualidade do produto final. A liga Pb-30%Sn com alto intervalo de solidificação em comparação com outras ligas testadas, apresentou maior dificuldade para ser tixolaminada. Isso ocorreu, porque as ligas de alto intervalo de solidificação tendem a formar trincas à quente no final da solidificação. Como resultado, uma pasta metálica plástica é difícil de formar. O caminho provável para obter uma tira semi-sólida fundida de boa qualidade neste caso, é aplicar uma inoculação que produz grãos finos antes do vazamento. O controle para a tixolaminação empregando a liga Sn-37%Pb com intervalo de solidificação menor, e elevada fluidez, é mais rigoroso para obter uma tira contínua. Conseqüentemente, foram utilizadas diferentes temperaturas de vazamento (260, 240 e 220 ºC) para controlar a fluidez e obter o tempo de contato... (Resumo completo, clicar acesso eletrônico abaixo) / Abstract: This is an experimental study of cast strip rolling from semi-solid material employing a cooling slope which continuously feeds a rolling mill. The cylinders of the rolling mill are positioned horizontally and can be operated at speeds of 0.25 m/s, 0.47 m/s, 0.73 m/s and 1.07 m/s. The lower speed of 0,25 m/s produces a strip of better quality. Hypoeutectic Pb/Sn alloys (Pb-30%Sn, Pb-40%Sn, Pb-50%Sn) and near eutectic point alloys (Pb-63%Sn), with solidification intervals of 75°C, 56°C, 31°C and 6°C respectively, according to the phase diagram, were used in experimental tests to obtain cast semi-solid and thixorolled strips for comparison. Simulations highlighted the necessary control parameters required to obtain good quality of the strip. These were: control alloy solidification interval, pouring temperature, roll speeds, ceramic nozzle temperature at the lower roll, quality of the roll surface finishing and tundish geometry. The Pb-30%Sn alloy, which has a much higher solidification interval in comparison with the other alloys tested, was difficult to thixoroll. This is because alloys with a high solidification interval tend to form hot tears at the end of solidification, and prevent a plastic metallic mush from forming. The probable solution to obtaining a semi-solid fused strip of good quality with this material, is to apply an inoculation that produces fine grains just before the pouring. In contrast, the parameter control for thixorolling of the Sn-37%Pb alloy, with lesser solidification interval and elevated fluidity, needed to be rigorous to obtain a continuous strip. Consequently, several pouring temperatures (260, 240 and 220ºC) were used to vary the fluidity and obtain sufficient alloy-inferior cylinder contact time for complete solidification. The strips obtained by the twin and single roll processing, and conventional rolling were characterized... (Complete abstract click electronic access below) / Mestre

This diploma thesis deals with the numerical modeling of the nail joint of the timber roof structure and it is based on the already realized physical experiment. Totally twelve various solutions, where the isotropic and ortotropic characteristics of the spruce timber and steel components change, have been created. The final values of the joint shift obtained by using of the numerical modeling are compared with results of physical experiment. Program system ANSYS is used.

The topic of this thesis is the numerical simulation of transversely isotropic 3D structures and laminates by means of the adaptive finite element method. To achieve this goal, the theoretical background of elastic deformation problems, transverse isotropy, plate theory, and the classical laminate theory is recapitulated. The classical laminate theory implies a combination of the membrane problem and the plate problem with additional coupling terms. The focus of this work is the adjustment of two integral parts of the adaptive FE algorithm according to the classical laminate theory. One of these parts is the solution of the FE system; a good preconditioner is needed in order to use the conjugate gradient method efficiently. It is shown via a spectral equivalence bound that the combination of existing preconditioners for the membrane and plate problems poses a capable preconditioner for the combined laminate problem. The other part is the error estimation process; the error estimator determines where the current mesh has to be refined for the next step. Existing results on residual error estimators for the elasticity problem, the biharmonic problem, and the plate problem are combined and extended to obtain a posteriori local residual error indicators for the classical laminate theory problem. The effectiveness of both results is demonstrated by numerical examples.:1 Introduction 1.1 Motivation 1.2 Organisation of this work 1.3 Notation and basic definitions 2 Basic theory of 3D simulation 2.1 Differential geometry 2.1.1 Initial and deformed domain 2.1.2 Strain tensor 2.2 Energy functional 2.2.1 Linearly elastic material law 2.2.2 Equilibrium of forces 2.2.3 Large deformations 2.2.4 Small deformations 2.3 Voigt notation and elasticity matrix 3 Transversely isotropic material law 3.1 Elasticity tensor 3.2 Conversion of the material constants 3.3 Elasticity matrix 3.4 Eigenvalues 3.5 State of plane strain 3.6 State of plane stress 4 Plate theory and classical laminate theory 4.1 The Kirchhoff–Love hypothesis 4.2 Constitutive law and bilinear form of the laminated plate 4.3 Definition of resultants 4.4 Boundary conditions 4.5 From the equilibrium conditions to the weak formulation 4.5.1 Membrane equilibrium 4.5.2 Plate equilibrium 4.5.3 Combined weak formulation 4.5.4 The CLT problem in Voigt notation 5 Discretisation 5.1 Short introduction to FEM 5.2 Adaptive FEM 5.3 Finite elements for 3D elasticity problems 5.4 Finite elements for plates 5.4 Finite elements for plates 5.4.1 BFS rectangles 5.4.2 rHCT triangles 5.5 CLT elements 5.5.1 Rectangles 5.5.2 Triangles 6 Solver and preconditioner 6.1 The preconditioned conjugate gradient method 6.2 Hierarchical basis and BPX preconditioners 6.3 Preconditioning of CLT problems 6.3.1 General laminates 6.3.2 Some special cases and examples 7 A posteriori residual error estimation 7.1 Residual error estimator for 3D elements 7.2 Residual error estimator for plate and CLT elements 7.2.1 Auxiliary definitions and assumptions on the mesh 7.2.2 Interpolation operators 7.2.3 Important inequalities 7.2.4 Cut-off functions 7.2.5 Definition of the error 7.2.6 Reliability inequality 7.2.7 Efficiency inequality 8 Some details of the implementation 8.1 The adaptive FE package SPC-PM 8.2 Remarks on some added features 8.2.1 Capability of the current code 8.2.2 Cuntze’s failure mode concept 8.3 Coordinate transformation of higher-order derivatives 8.3.1 Mapping of coordinates 8.3.2 Transformation of derivatives of up to the third-order 8.3.3 Recursive construction of transformation matrices 8.3.4 Simplification for axis-parallel rectangles 9 Numerical examples 9.1 A three-dimensional example from eniPROD 9.2 Example problems for laminates 9.2.1 Rectangular plate under in-plane load 9.2.2 Rectangular plate under vertical load 9.2.3 L-shaped plate with inhomogeneous natural boundary conditions 10 Conclusion and outlook Bibliography Acknowledgements List of main symbols Theses