This thesis presents the implementation of the upper bound limit analysis in combination with finite elements and linear optimization for solving different stability problems in geomechanics under plane strain conditions. Although the nonlinear optimization techniques are becoming quite popular, the linear optimization has been adopted due to its simplicity in implementation and ease in attaining the convergence while performing the analysis. The objectives of the present research work are (i) to reduce the computational effort while using an upper bound finite element limit analysis with linear programming in dealing with geotechnical stability problems, and (ii) to obtain solutions for a few important geotechnical stability problems associated with reinforced earth, unsupported tunnels and a group of anchors. It is also intended to examine the developments of the failure patterns in all the cases. For carrying out the analysis for different stability problems, three noded triangular elements have been used throughout the thesis. The nodal velocities are treated as basic unknown variables and the velocity discontinuities are employed along the interfaces of all the elements. The soil mass is assumed to obey the Mohr-Coulomb’s failure criterion and an associated flow rule. The Mohr-Coulomb yield surface is linearized by means of an exterior regular polygon circumscribing the actual yield circle so that the finite element formulation leads to a linear programming problem. A simple technique has been proposed for reducing the computational effort while solving any geotechnical stability problem by using the upper bound finite element limit analysis and linear optimization. In the proposed method, the problem domain has been discretized into a number of different regions in which a particular order (number of sides) of the polygon has been specified to linearize the Mohr-Coulomb yield criterion. A greater order of the polygon needs to be chosen only in that part of the domain wherein the rate of the plastic strains becomes higher. The computational effort required to solve the problem with this implementation reduces considerably. By using the proposed method, the bearing capacity has been computed for smooth as well as rough strip footings and the results obtained are found to be quite satisfactory. The ultimate bearing capacity of a rigid strip footing placed over granular, cohesive-frictional and purely cohesive soils, reinforced with single and a group of two horizontal layers of reinforcements has been determined. The necessary formulation has been introduced to incorporate the inclusion of reinforcement in the analysis. The efficiency factors, and , to be multiplied with Nc and Nγ for finding the bearing capacity of reinforced foundations, have been established. The results have been obtained (i) for different values of soil friction angles in case of granular and cohesive-frictional soils, and (ii) for different rates at which the cohesion increases with depth for purely cohesive soil under undrained condition. The optimum positions of the reinforcements' layers corresponding to which and becomes maximum, have been established. The effect of the length of the reinforcements on the results has also been analyzed. As compared to cohesive soil, the granular soils, especially with greater values of frictional angle, cause much more predominant increase in the bearing capacity. The stability of a long open vertical trench laid in a fully cohesive and cohesive-frictional soil has been determined with an inclusion of single and a group of two layers of horizontal reinforcements. For different positions of the reinforcement layers, the efficiency factor (ηs), has been determined for several combinations of H/B, m and where H and B refer to height and width of the trench, respectively, and m accounts for the rate at which the cohesion increases linearly with depth for a fully cohesive soil with = 0. The effect of height to width of the long vertical trench on the stability number has been examined for both unreinforced and reinforced soils. The optimal positions of the reinforcements layers, corresponding to which becomes maximum, have been established. The required length of reinforcements to achieve maximum efficiency factor corresponding to optimum depth of reinforcement has also been determined. The magnitude of the maximum efficiency factor increases continuously with an increase in both m and . The effect of pseudo-static horizontal earthquake body forces on the stability of a long unsupported circular tunnel (opening) formed in a cohesive frictional soil has been determined. The stability numbers have been obtained for various values of H/D (H = tunnel cover, D = diameter of the tunnel), internal friction angle of soil, and the horizontal earthquake acceleration coefficient The computations revealed that the values of the stability numbers (i) decreases quite significantly with an increase in , and (ii) become continuously higher for greater values of H/D and . The failure patterns have also been drawn for different combinations of H/D, and . The geometry of the failure zone around the periphery of the tunnel becomes always asymmetrical with an inclusion of horizontal seismic body forces. The interference effect on the stability of two closely spaced parallel (twin) long unsupported circular tunnels formed in fully cohesive and cohesive-frictional soils has been evaluated. The variation of the stability number with S/D has been established for different combinations of H/D, m and ; where D refers to the diameter of each tunnel, S is the clear spacing between the tunnels, and is the internal friction angle of soil and m accounts for the rate at which the cohesion increases linearly with depth for a soil with = 0. On account of the interference of two tunnels, the stability number reduces continuously with a decrease in the spacing between the tunnels. The minimum spacing between the two tunnels required to eliminate the interference effect increases with (i) an increase in H/D and (ii) a decrease in the values of both m and . The failure patterns have also been generated for a few cases with different values of S/D. The size of the failure zone is found to become smaller for greater values of m and . The horizontal pullout capacity of a group of two vertical strip anchors embedded, along the same vertical plane in sand, at shallow depths has been determined. At collapse, it is assumed that the anchor plates are subjected to the same uniform horizontal velocity without any bending or tilt. The pullout resistance increases invariably with increases in the values of embedment ratio, friction angle of the sand mass and anchor-soil interface friction angle. The effect of spacing (S) between the anchors on their group collapse load is examined in detail. For a given embedment ratio, the total group failure load becomes maximum corresponding to a certain optimal spacing (Sopt). The values of Sopt increases with an increase in the value of , but the changes in the value of H/B and do not have any significant effect on Sopt. The vertical uplift capacity of a group of two horizontal strip plate anchors with the common vertical axis buried in purely cohesive as well as in cohesive frictional soil has been computed. The variation of the uplift factors Fc, Fq and F , due to the contributions of soil cohesion, surcharge pressure and unit weight, respectively, has been evaluated for different combinations of S/B and H/B. As compared to a single isolated anchor, the group of two anchors generates significantly greater magnitude of Fc. On the other hand, the factors Fq and F , for a group of two anchors are found to become almost equal to that of a single isolated anchor as long as the levels of the lower plate in the group and the single isolated anchor are kept the same. For the group of two horizontal strip plate anchors in purely cohesive soil, an increase of cohesion of soil mass with depth and the effect of self weight of the soil have been incorporated. The uplift factor Fcy both due to cohesion and unit weight of the soil has also been computed for the anchors embedded in clay under undrained condition. For given embedment ratios, the factor Fcy increases linearly with an increase in the normalized unit weight of soil mass upto a certain value before attaining a certain maximum magnitude. The computational results obtained for different research problems would be useful for design.

Shock-shock interactions are among the most basic gas-dynamic problem, and are almost unavoidable in any high speed light, where shock waves generating from different sources crosses each other paths. These interactions when present very close to the solid surface lead to very high pressure and thermal loads on the surface. The related practical problem is that experienced at the cowl lip of a scramjet engine, where the interfering shock waves leads to high heat transfer rates which may also lead to the damage of the material. The classification by Edney (1968) on the shock-shock interaction patterns based on the visualization has since then served the basis for such studies. Though the problem of high heating on the surface in the vicinity of the shock-shock interactions has been studied at length at supersonic Mach numbers, the study on the topic at the hypersonic Mach numbers is little sparse. Even in the studies at hypersonic Mach numbers, the high speeds are not simulated, which is the measure of the kinetic energy of the ow. Very few experimental studies have addressed this problem by simulating the energy content of the ow. Also, some of the numerical studies on the shock-shock interactions suggest the presence of unsteadiness in the shock-shock interaction patterns as observed by Edney (1968), though this observation is not made very clearly in the experimental studies undertaken so far. In the present study, experiments are carried out in a conventional shock tunnel at Mach number of 5.62 (total enthalpy of 1.07 MJ/kg; freestream velocity of 1361 m/s), with the objective of mapping the surface pressure distribution and surface convective heat transfer rate distribution on the hemispherical body in the presence of the shock-shock interactions. A shock generator which is basically a wedge of angle = 25 , is placed at some dis-dance in front of the hemispherical body such that the planar oblique shock wave from the shock generator hits the bow shock wave in front of the hemi-spherical body. The relative distance between the wedge tip and the nose of the hemispherical body is allowed to change in di erent experiments to capture the whole realm of shock-shock interaction by making the planar oblique shock wave interact with the bow shock wave at different locations along its trajectory. The study results in a bulk of data for the surface pressure and heat transfer rates which were obtained by placing 5 kulites pressure transducers, 1 PCB pressure transducer and 21 platinum thin lm gauges along the surface of the hemispherical body in a plane normal to the freestream velocity direction. Along with the measurement of the surface pressure and the surface heat transfer rates, the schlieren visualization is carried out to capture the shock waves, expansion fans, slip lines, present in a certain shock-shock interaction pattern and the measured values were correlated with the captured schlieren images to evaluate the ow build up and steady and useful test time thereby helping in understanding the ow physics in the presence of the shock-shock interactions. From the present study it has been observed that in the presence of Edney Type-I and Edney Type-II interaction, the heat transfer rates on the hemi-spherical body are symmetrical about the centerline of the body, with the peak heating at the centerline which drops towards the shoulder. For Edney Type-III, Edney Type-IV, Edney Type-V and Edney Type-VI interaction pattern, the distribution in not symmetrical and shifts in peak heat transfer rates being on the side of the hemispherical from which planar oblique shock wave is incident. Also, it is observed that for the interactions which appear within the sonic circle, Edney Type-III and Edney Type-IV, the heat transfer rates observe an unsteadiness, such that the gauges located close to the interaction region experiencing varying heat transfer rates during the useful test time of the shock tunnel. Few experiments were conducted at Mach 8.36 (total enthalpy of 1.29 MJ/kg; freestream velocity of 1555.25 m/s) and Mach 10.14 (total enthalpy of 2.67 MJ/kg; freestream velocity of 2258.51 m/s) for the con gurations representing Edney Type-III interaction pattern to further evaluate the unsteady nature observed at Mach 5.62 ows. The unsteadiness was evident in both the cases. It is realized that the short test times in the shock tunnels pose a constraint in the study of unsteady flow fields, and the use of tailored mode operation of shock tunnel can alleviate this constraint. Also, limited number of experiments in the present study, which are carried out in a Free Piston Shock Tunnel, helps to understand the need to conduct such study in high enthalpy test conditions.

published_or_final_version / Urban Planning / Master / Master of Science in Urban Planning

published_or_final_version / Business Administration / Master / Master of Business Administration

Le travail présenté dans cette thèse s'inscrit à la fois dans le domaine de l'informatique musicale et dans celui de la réalité virtuelle. Nous présentons tout d'abord les avantages des environnements virtuels immersifs et de l'interaction graphique 3D pour les instruments multi-processus et en particulier pour le live-looping hiérarchique, un nouveau modèle d'instrument que nous définissons. Nous proposons ensuite des solutions pour la visualisation et la manipulation des processus sonores dans ces environnements virtuels. Afin de représenter les processus sonores, nous introduisons les widgets réactifs 3D qui s'appuient sur des mappings audiovisuels que nous évaluons. Nous développons un nouveau dispositif d'interaction 3D adapté à l'interaction musicale, appelé Piivert, ainsi que des techniques associées, dont les tunnels afin de manipuler les widgets réactifs 3D. Finalement, nous mettons en œuvre le live-looping hiérarchique avec l'instrument immersif Drile, en utilisant les techniques développées précédemment, et nous étudions plusieurs scènes pour les performances musicales immersives.

Cette thèse s'inscrit dans la continuation du travail de Younes Achkire, consacré au contrôle actif des ponts haubanés. Elle traite de l'implémentation d'un système de contrôle actif sur des maquettes de structures câblées. Deux types de structures sont étudiés expérimentalement: les ponts haubanés et les treillis spatiaux. Après une brève introduction sur l'usage du contrôle actif dans ces domaines, le chapitre 2 traite numériquement des mécanismes d'interaction entre le câble et la structure. Au chapitre 3, nous présentons la stratégie de contrôle que nous utilisons pour stabiliser une structure câblée: il s'agit d'un contrôle décentralisé, basé sur des paires capteur/actionneur colocalisées, placées au niveau des ancrages des câbles, chacune équipée d'un contrôleur Intégral Force Feedback. Nous présentons une théorie linéaire simplifiée permettant de dimensionner le système et de prévoir son efficacité. Elle est illustrée sur un exemple, et nous discutons de la validité de certaines hypothèses simplificatrices. Le chapitre 4 est consacré au contrôle actif des ponts haubanés. Nous y présentons 2 maquettes. La première, de petite taille (3m) représente un pylône de pont haubané en construction. Elle est équipée d'actionneurs piézoélectriques. La seconde, installée au Centre Commun de Recherche d'Ispra (Italie), mesure 30m de long, et est équipée d'actionneurs hydrauliques. Les expériences réalisées sur ces maquettes ont démontré l'efficacité du contrôle et la fiabilité de la théorie prédictive. Le contrôle du flottement des ponts est traité sur un exemple numérique. Le chapitre 5 relate nos expériences d'amortissement actif des treillis spatiaux. Deux structures ont été étudiées: une colonne en treillis équipée de 3 câbles actifs, et une structure triédrique suspendue à des cordons élastiques pour simuler l'absence de gravité, également munie de câbles actifs. Deux concepts d'actionneur piézoélectrique ont été testés. Nous avons ensuite examiné le problème de la saturation des actionneurs, et celui du contrôle actif des microvibrations (~10nm) d'une structure câblée. Le chapitre 6 conclut ce travail, en souligne les aspects originaux et donne quelques perspectives de développement.

Recently, the wavy leading edge airfoils, inspired by the humpback whale´s flipper, have been investigated, as flow control mechanisms, at low Reynolds numbers in order to improve aerodynamic performance in this particular flow regime. The overall aim of this work is to investigate the airfoil geometric effects on wavy leading edge phenomena in the low Reynolds number regime. Experimental investigations were carried out correlating force measurements with mini-tuft and oil visualizations in order to understand the airfoil thickness effects on wavy leading edge phenomena. Three sets of airfoil thickness were tested (NACA 0012, NACA 0020 and NACA 0030), each set consisting of smooth plus three wavy configurations (A=0.11c, ?=0.40c; A=0.03c, ?=0.40c and A=0.03c, ?=0.11c); Reynolds number was varied between 50,000 and 290,000. The results present many findings that were not possible in previous studies due the fact that these investigations were constrained to specific geometries and/or flow conditions. At higher Reynolds number, the decrease in airfoil thickness leads the airfoils to leading edge stall characteristics causing the lowest aerodynamic deterioration for the thinnest wavy airfoil as compared to smooth configuration in the pre-stall regime. In addition, the results show impressive tubercle performance in the lowest Reynolds number. For any tubercle geometry and airfoil thickness, the wavy leading edge airfoils present higher maximum lift values as compared to smooth configurations showing an unprecedented increase in performance for a full-span model tested in the literature. The flow visualizations present two flow mechanisms triggered by secondary flow: three-dimensional laminar separation bubbles and vortical structures. Regarding three-dimensional laminar bubbles, the results confirm some of the few previous experimental and numerical studies, and presents for the first time these structures as a very efficient flow control mechanism in the post-stall regime justifying the impressive increase in maximum lift in the lowest Reynolds number. Besides that, two characteristics of laminar bubbles, \"tipped-bubbles\" and \"elongated-bubbles\", are identified with different effects in the pre-stall regime. This thesis presents higher tubercle performance for thinner airfoils (NACA 0012) and/or lower Reynolds number conditions (Re=50,000) showing clearly that an optimum performance lead the \"tubercles\" to operate under conditions of leading edge flow separation conditions. Therefore, a design space for tubercles conducted to leading edge stall characteristics confirming the hypothesis of Stanway (2008) eight years before. / Recentemente, aerofólios com bordo de ataque ondulados, inspirados na nadadeira da baleia jubarte, tem sido investigados como mecanismo de controle de escoamento para baixo número de Reynolds com a finalidade de se aumentar o desempenho aerodinâmico neste específico regime de escoamento. O objetivo geral deste trabalho é investigar os efeitos geométricos do aerofólio nos fenômenos do bordo de ataque ondulado na condição de baixo número de Reynolds. Investigações experimentais foram realizadas correlacionando medições de forças com visualizações de lã e óleo a fim de compreender os efeitos da espessura do aerofólio sobre os fenômenos de bordo de ataque ondulado. Três conjuntos de espessura de aerofólios foram testados (NACA 0012, NACA 0020 e NACA 0030) na faixa de número de Reynolds entre 50,000 e 290,000, onde cada conjunto tem um aerofólio liso e três ondulados (A = 0.11c, ? = 0.40c; A = 0.03c, ? = 0.40c e A = 0.03c, 0.11c ? =0.11c). O dados experimentais mostram importantes resultados que não foram possíveis em estudos anteriores devido às investigações serem restritas à geometria ou/e condição de escoamento específicas. O resultados de medida de força mostram que a diminuição da espessura do aerofólio conduz às características de separação de escoamento de bordo de ataque que causam menor deterioração aerodinâmica nos aerofólios ondulados finos quando comparados aos lisos no regime de pré-stall. Além disso, os resultados mostram um desempenho destacado do bordo de ataque ondulado para condição de menor número de Reynolds. Em quaisquer espessuras de aerofólio, os bordos ondulados apresentam valores de sustentação máxima maiores quando comparado aos aerofólios lisos mostrando assim resultado inédito na literatura para modelos ondulados bi-dimensionais. As visualizações de óleo evidenciaram dois mecanismos de controle de escoamento desencadeadas pelo escoamento secundário: bolhas de separação laminar tridimensionais e estruturas vorticais. Os resultados confirmam alguns poucos estudos experimentais e numéricos anteriores relacionadas com bolhas tridimensionais, e apresenta pela primeira vez estas estruturas como um mecanismo muito eficiente de controle de escoamento em regime de pós-stall justificando o aumento de máxima sustentação para o menor número de Reynolds. Adicionalmente, foram identificadas duas estruturas de bolhas tridimensionais nomeados aqui como \"bolhas com pontas\" e \"bolhas alongadas\" que causam distintos efeitos no regime de pré-stall. Esta tese apresenta como resultado maior desempenho para aerofólios ondulados com menor espessura (NACA 0012) e/ou para condições de menor número de Reynolds (Re=50,000)mostrando claramente que estas características levam as ondulações a operarem em condições de stall de bordo de ataque assim tendo um desempenho superior. Portanto, um espaço de projeto para tubérculos conduz às características de stall de bordo de ataque confirmando a suposição de Stanway (2008) oitos anos antes.

O presente trabalho aborda o estudo do comportamento de túneis em maciços de solo. É dada ênfase na aplicação de análises numéricas com emprego de diferentes modelos constitutivos elasto-plásticos para solos. São apresentadas análises numéricas tridimensionais de um túnel amplamente instrumentado pertencente à Companhia do Metropolitano de São Paulo. As análises são realizadas com auxílio de um programa computacional de elementos finitos. O comportamento do maciço em pontos situados ao redor da escavação é minuciosamente estudado e a capacidade dos modelos em representar adequadamente o comportamento verificado na obra pelas instrumentações é avaliada. Além das análises numéricas o trabalho aborda os principais conceitos relacionados com escavações de túneis em maciços de solo. Conceitos relacionados com a engenharia prática de túneis são apresentados de maneira qualitativa, sem formulações teóricas e matemáticas. Uma revisão bibliográfica com publicações recentes das principais revistas e periódicos que tratam do tema de análise numérica aplicada a túneis é apresentada. O trabalho também apresenta uma revisão dos principais conceitos relacionados com os modelos constitutivos comumente utilizados para análise de problemas de geotecnia. Além do modelo elástico são apresentados os modelos elasto-plásticos de Tresca, von Misses, Drucker-Prager e Mohr-Coulomb. Uma breve introdução aos conceitos básicos de estado crítico, juntamente com as equações do modelo Cam-Clay original e Cam-Clay modificado são apresentadas. Antes da apresentação das equações desses modelos constitutivos, são introduzidos os conceitos básicos relacionados com o comportamento dos materiais elasto-plásticos. Os conceitos de material elástico perfeitamente plástico e de material com endurecimento (ou hardening) e amolecimento (ou softening) são apresentados. Os conceitos de superfície de plastificação e de superfície de potencial plástico também são apresentados. Por fim, são sintetizados os pontos mais relevantes da pesquisa realizada, apontando as limitações do trabalho com sugestões de novos estudos a serem realizados nessa mesma linha de pesquisa. / The present research approaches the study of the behaviour of tunnels in soil. It is given emphasis in the application of numerical analyses using different elasto-plastic constitutive models for soils. Three-dimensional numerical analyses of a widely instrumented tunnel belonging to the Company of the Metropolitan of São Paulo are presented. The analyses are carried through with aid of a computational program of finite elements. The behaviour of the soil mass in points located around the excavation is thoroughly studied and the capacity of the models in adequately representing the field behavior verified by the instrumentations is evaluated. Furthermore, the work approaches the main concepts related to tunneling in soils. Concepts related to practical engineering of tunnels are presented in a qualitative way, without theoretical and mathematical formulations. A literature review of recent publications of the most important periodic magazines and that deal with the subject of numerical analysis applied to tunnels is presented. The work also presents a revision of the main concepts related to the constitutive models normally used for analysis of geotechnical problems. Beyond the elastic model the elasto-plastics models of Tresca, von Misses, Drucker-Prager and Mohr-Coulomb are presented. Brief introductions to the basic concepts of critical state, together with the equations of the (original) Cam-Clay original and (modified) Cam-Clay modified models are presented. Before the presentation of the equations of these constitutive models, the basic concepts of the behaviour of the elasto-plastics materials are introduced. The concepts of perfectly plastic elastic material and material with hardening and softening are presented. The concepts of plastic surface and plastic potencial surface are also presented. Finally, the most relevant points of the research are synthesized, pointing the limitations of the developed work along with suggestions for new studies to be carried through in this line of research.

Os sistemas de classificação maciços rochosos e as redes neurais artificiais possuem diversas similaridades. Existem características que estão presentes nos dois tipos de sistemas: bases de dados são usadas para o seu desenvolvimento; e pesos são parte da representação do conhecimento. Os principais sistemas de classificação geomecânicas (Sistema Q e RMR) podem ser escritos como representações neurais locais. Tais representações permitem uma melhor compreensão do processo de classificação e identificação de padrões realizado pelas classificações convencionais. Experimentos convencionais foram realizados com modelos de redes neurais não-supervisionados. Os modelos não supervisionados permitiriam uma melhor compreensão da distribuição dos dados no espaço de feições. Um modelo supervisionado para escavações subterrâneas em todo domínio do espaço de feições. Importantes relações entre características foram encontradas. / Rock mass classification systems and artificial neural networks have several similarities. There is some characteristics present in both systems: data bases are used in they development, and weights are part of the knowledge representation. The main rock mass classification systems (Q-system and RMR) can be written as local neural network representations. This representation helps a better understanding of the pattern classification and identification process made by the conventional classifications. Computational experiments were made with unsupervised and supervised neural networks models. Unsupervised models allow a better understanding of the data in the feature space. A supervised model allow to make a mapping of the support type used in underground excavation in all feature space domain. Important relations between domain regions characteristics and type of support used were found.

[pt] Nesse trabalho é avaliada a solução numérica do colapso na frente de escavação em túneis rasos, através da teoria de análise limite numérico, usando o teorema do limite inferior, a partir da condição de equilíbrio para as condições plásticas, além de considerar o comportamento do material rígido perfeitamente plástico. O teorema de limite inferior implica em maximizar o fator multiplicador na carga atuante, por isso a análise limite se torna um problema de otimização, nele tem que se usar a programação matemática para ser resolvido. É avaliada a solução numérica tridimensional da análise limite através do método dos elementos finitos, usando malha de elementos hexaédricos de oito nós, a análise dos elementos finitos é feita com o próprio código gerado na linguagem de programação do MATLAB 2017.As metodologias de programação matemática empregadas são: programação cônica de segunda ordem e programação semidefinida. Antes deve-se adaptar os critérios de ruptura de Drucker Prager à programação cônica de segunda ordem e Mohr-Coulomb tridimensional à programação semidefinida. Para a otimização se usa o algoritmo comercial MOSEK Aps 7.1 baseado no método do ponto interior em grande escala, na linguagem do MATLAB 2017. Além disso, obteve-se o mecanismo de colapso através da propriedade da dualidade do problema de otimização, dualidade que é cumprida pelos teoremas de limite superior e inferior. / [en] In this work the numerical solution of the collapse in the front of excavation in shallow tunnels is evaluated through the theory of numerical limit analysis, using the lower limit theorem, from the equilibrium condition for the plastic conditions, considering the behavior of the perfectly plastic rigid material. The lower limit theorem implies maximizing the multiplier factor in the acting load, so that the limit analysis becomes an optimization problem. The three-dimensional numerical solution of the limit analysis using the finite element method is evaluated using a mesh of eight-node hexahedral elements. The finite element analysis is done using the code generated in the MATLAB 2017 programming language. The mathematical programming methodologies used are: second order conic programming and semidefinite programming. The Drucker-Prager three-dimensional criteria should be adapted to the conic programming of the second order and Mohr-Coulomb three-dimensional to the semidefinite programming. For the optimization, the MOSEK Aps 7.1 commercial algorithm based on the large-scale interior point method is used in the MATLAB 2017 language. In addition, the collapse mechanism was obtained through the duality property of the optimization problem, duality that is fulfilled by the upper and lower limit theorems.