Hydrogen fuel cells are devices used to generate electricity from the electrochemical reaction between air and hydrogen gas. An attractive advantage of these devices is that their byproduct is water, which is very safe to the environment. However, hydrogen fuel cells still lack some improvements in terms of increasing their life time and electricity production, decreasing power losses, and optimizing their operating conditions. In this thesis, the cathode part of the hydrogen fuel cell will be considered. This part mainly consists of an air gas channel and a gas diffusion layer. To simulate the fluid dynamics taking place in the cathode, we present two models, a general model and a simple model both based on a set of conservation laws governing the fluid dynamics and chemical reactions. A numerical method to solve these models is presented and verified in terms of accuracy. We also show that both models give similar results and validate the simple model by recovering a polarization curve obtained experimentally. Next, a shape optimization problem is introduced to find an optimal design of the air gas channel. This problem is defined from the simple model and a cost functional, $E$, that measures efficiency factors. The objective of this functional is to maximize the electricity production, uniformize the reaction rate in the catalytic layer and minimize the pressure drop in the gas channel. The impact of the gas channel shape optimization is investigated with a series of test cases in long and short fuel cell geometries. In most instances, the optimal design improves efficiency in on- and off-design operating conditions by shifting the polarization curve vertically and to the right. The second primary goal of the thesis is to analyze mathematical issues related to the introduced shape optimization problem. This involves existence and uniqueness of the solution for the presented model and differentiability of the state variables with respect to the domain of the air channel. The optimization problem is solved using the gradient method, and hence the gradient of $E$ must be found. The gradient of $E$ is obtained by introducing an adjoint system of equations, which is coupled with the state problem, namely the simple model of the fuel cell. The existence and uniqueness of the solution for the adjoint system is shown, and the shape differentiability of the cost functional $E$ is proved.

Randomised controlled trials (RCT) are considered the gold standard for assessing the efficacy and safety of medical interventions. However, RCTs face unique challenges when applied to medical technologies, such as issues related to timing of assessment, eligible population, acceptability, blinding, choice of comparator group, and consideration for learning curves. To address these challenges, this thesis explores the adaptation of the model-informed drug development (MIDD) approach to the field of medical technology, using a case study on transurethral microwave thermotherapy (TUMT). The research employs non-linear mixed- effects (NLME) modelling and D-optimal design to optimise study designs and improve the reliability and efficiency of clinical trials. The impact of different sampling times, sample sizes, and learning curves on effect size estimates is analysed. The results show that optimising sampling points and sizes significantly improves the precision and reliability of effect size estimates and describes how MIDD can be a useful tool for this purpose. The study also highlights the limitations of the TUMT study design, suggesting ways in which the model-based approach could offer more robust and reliable clinical evidence generation. This research highlights the potential of the MIDD approach to streamline the medical technology clinical development process, enhance the quality of evidence, and address its inherent complexities. Future work should expand on these findings by exploring more complex error models and additional study designs and its related aspects. / Randomiserade kontrollerade studier (RCT) anses vara standard för att bedöma effekt och säkerhet i kliniska interventionsstudier. RCT:er står dock inför unika utmaningar när de tillämpas på medicinteknik såsom utmaningar relaterade till tidpunkt för bedömning, rekrytering av lämpliga studiedeltagare, acceptans, blindning, val av jämförelsegrupp och hänsyn till inlärningskurvor. För att hantera dessa utmaningar undersöker denna avhandling anpassningen av modellinformerad läkemedelsutveckling (MIDD) till området medicinteknik, med hjälp av en fallstudie om transuretral mikrovågstermoterapi (TUMT). I arbetet tillämpas icke-linjär, hierarkisk (NLME) modellering och D-optimal design för att optimera studiedesigner och förbättra tillförlitligheten i kliniska prövningar. Effekten av olika observationstider, antal studiedeltagare och inlärningskurvor på estimeringen av effektstorlek analyseras. Resultaten visar att optimering av observationstidpunkter och studiestorlek avsevärt förbättrar precisionen och tillförlitligheten av den estimerade effektstorleken och visar på hur MIDD kan vara ett användbart verktyg för detta ändamål inom medicinteknisk utveckling. Studien belyser också begränsningarna i studiedesignen för fallstudien och föreslår hur en modellbaserad metod skulle kunna erbjuda mer robust och tillförlitlig generering av klinisk evidens. Denna forskning belyser potentialen hos MIDD-metoder för att effektivisera den medicintekniska kliniska utvecklingsprocessen, förbättra kvaliteten av evidens, och hantera dess inneboende komplexitet. Framtida arbete bör utvidga dessa resultat genom att utforska mer komplexa modeller, ytterligare studiedesigner, och relaterade aspekter.

Darbo tikslas – tampriųjų-plastinių strypų konstrukcijų (santvaros, rėmų), veikiamų kartotinės kintamosios apkrovos, prisitaikomumo proceso ir būvio įtempių ir deformacijų analizė optimalaus konstrukcijų projektavimo kontekste. Darbo aktualumas grindžiamas prisitaikančių konstrukcijų optimizavimo uždaviniuose figūruojančių standumo sąlygų-apribojimų kokybės gerinimo būtinumu. Prisitaikančių konstrukcijų deformacijų būvis priklauso nuo apkrovimo istorijos, o poslinkių ribojimui taikoma nepakankamai tiksli Koiterio sąlyga arba liekamųjų poslinkių influentinė matrica, nepagrįstai laikant, kad plastinio deformavimo procesas yra išimtinai holonominis. Anotuojamame darbe apkrovimo istoriją siūloma įvertinti, atliekant papildomą inkrementinę deformacijų būvio analizę. Tyrimai atlikti taikant idealiai tamprių plastinių santvarų ir rėmų techniškosios skaičiavimo teorijos prielaidas (maži poslinkiai ir deformacijos). Taikomi ekstreminiai energiniai mechanikos principai, matematinio programavimo teorija ir metodai. Inkrementinės analizės matematiniai modeliai sudaryti, besikeičiančias plastines deformacijas tapatinant su distorsijomis. Taip nustatomos konkrečios apkrovimo istorijos liekamųjų poslinkių kitimo maksimalios ir minimalios reikšmės. Gautieji rezultatai panaudoti optimizavimo uždavinių sprendiniams tikslinti ir, esant būtinumui, leidžiantys keisti pradines pagrindinio optimizavimo uždavinio sąlygas. Pateikti išsamūs skaitinių eksperimentų rezultatai. / The purpose of this work is analysis of stress-deformation state of perfectly elastic-plastic shakedown structures (truss, frames) subjected to repeated variable load in the context of optimal design. Relevance of this work is based on a need of improvement of accurateness of stiffness constrains in optimization problems of structures. Stress-deformation state of shakedown structures depends on its loading history, while for the restriction of displacements inaccurate Koiter’s condition or an influence matrix of residual displacements, on the wrong supposition that the process of plastic deformation is exclusively holonomic, is used. In this work is proposed to evaluate loading history by performing an additional incremental analysis of stress-deformation state. This research was performed invoking the assumptions of technical computing theory of perfectly elastic-plastic trusses and frames (small deformations and displacements). Mechanics extremum energy principles, mathematical programming theory and methods are applied. Mathematical models of incremental analysis are composed by indentifying volatile plastic deformations with distortions. Thus particular maximum and minimum values of residual displacements are found. Obtained results are used to verify optimal design problems solutions and change the restrictions of main optimimization problem if necessary. Comprehensive results of numerical experiments presented.

The costs of developing new pharmaceuticals have increased dramatically during the past decades. Contributing to these increased expenses are the increasingly extensive and more complex clinical trials required to generate sufficient evidence regarding the safety and efficacy of the drugs.  It is therefore of great importance to improve the effectiveness of the clinical phases by increasing the information gained throughout the process so the correct decision may be made as early as possible.   Optimal Design (OD) methodology using the Fisher Information Matrix (FIM) based on Nonlinear Mixed Effect Models (NLMEM) has been proven to serve as a useful tool for making more informed decisions throughout the clinical investigation. The calculation of the FIM for NLMEM does however lack an analytic solution and is commonly approximated by linearization of the NLMEM. Furthermore, two structural assumptions of the FIM is available; a full FIM and a block-diagonal FIM which assumes that the fixed effects are independent of the random effects in the NLMEM. Once the FIM has been derived, it can be transformed into a scalar optimality criterion for comparing designs. The optimality criterion may be considered local, if the criterion is based on singe point values of the parameters or global (robust), where the criterion is formed for a prior distribution of the parameters.  Regardless of design criterion, FIM approximation or structural assumption, the design will be based on the prior information regarding the model and parameters, and is thus sensitive to misspecification in the design stage.  Model based adaptive optimal design (MBAOD) has however been shown to be less sensitive to misspecification in the design stage.   The aim of this thesis is to further the understanding and practicality when performing standard and MBAOD. This is to be achieved by: (i) investigating how two common FIM approximations and the structural assumptions may affect the optimized design, (ii) reducing runtimes complex design optimization by implementing a low level parallelization of the FIM calculation, (iii) further develop and demonstrate a framework for performing MBAOD, (vi) and investigate the potential advantages of using a global optimality criterion in the already robust MBAOD.

En France, il existe deux modes de traction pour les trains : la traction diesel ou la traction électrique. Chaque mode fait face à des problématiques qui lui sont propres. Dans le cas du diesel, les émissions de gaz à effet de serre étant de plus en plus contrôlées, il devient nécessaire de faire évoluer ce type de train vers une solution moins polluante. Dans le cas de la traction électrique, la consommation d’énergie entraine une chute de tension qui peut imposer un ralentissement des trains, empêchant ainsi le développement du trafic. La solution étudiée par la SNCF est l’hybridation des trains (ajout de systèmes de stockage en embarqué).Ces travaux de thèse ont pour objectif de mettre en place une méthode permettant de faire le pré-dimensionnement des systèmes de stockage embarqués dans le train. De plus, afin de tenir compte de l’influence réciproque de la gestion sur le dimensionnement, celle-ci est incluse dans le modèle de dimensionnement. La résolution du modèle global se fait à l’aide d’un algorithme d’optimisation.La méthode a été mise en place sur les deux modes de traction ferroviaire (diesel et électrique) et l’optimisation a été faite avec l’algorithme SQP (Sequential Quadratic Programming). / In France, there are two traction modes for railway: the diesel and electric traction. Each mode has its own issues. For diesel, the increasing control of the greenhouse gas emissions imposes to evolve this type of train to a less polluting solution. For electric traction, the energy consumption creates a voltage drop which can cause a traffic slowdown, which will limit the traffic development. The studied solution by SNCF is the hybridization of the train (adding storage system).Thus, these works have the objective to build a method to do the pre-sizing of storage systems embedded in trains. Moreover, to take into account the mutual influence of the sizing and the energy management, this last one is included in the sizing model. An optimization algorithm solves the global model.The method has been developed for the two traction modes (diesel and electric) and the optimization has been made with SQP algorithm (Sequential Quadratic Programming).

<p>Forest quantification methods have evolved from a simple graphical approach to complex regression models with stochastic structural components. Currently, mixed effects models methodology is receiving attention in the forestry literature. However, the review work (Paper I) indicates a tendency to overlook appropriate covariance structures in the NLME modeling process.</p><p>A nonlinear mixed effects modeling process is demonstrated in Paper II using Cupressus lustanica tree merchantable volume data and compared several models with and without covariance structures. For simplicity and clarity of the nonlinear mixed effects modeling, four phases of modeling were introduced. The nonlinear mixed effects model for C. lustanica tree merchantable volume with the covariance structures for both the random effects and within group errors has shown a significant improvement over the model with simplified covariance matrix. However, this statistical significance has little to explain in the prediction performance of the model.</p><p>In Paper III, using several performance indicator statistics, tree taper models were compared in an effort to propose the best model for the forest management and planning purpose of the C. lustanica plantations. Kozak's (1988) tree taper model was found to be the best for estimating C. lustanica taper profile.</p><p>Based on the Kozak (1988) tree taper model, a Ds optimal experimental design study is carried out in Paper IV. In this study, a Ds-optimal (sub) replication free design is suggested for the Kozak (1988) tree taper model.</p>

Current status data are usually obtained with a failure time variable T which is diffcult observed but can be determined to lie below or above a random monitoring time or inspection time t. In this work we consider bivariate current status data ${t,delta_1,delta_2}$ and assume we have some prior information of the bivariate failure time variables T1 and T2. Our main goal is to find an optimal inspection time for testing the relationship between T1 and T2.

Bivariate or multivariate correlated data may be collected on a sample of unit in many applications. When the experimenters concern about the failure times of two related subjects for example paired organs or two chronic diseases, the bivariate binary data is often acquired. This type of data consists of a observation point x and indicators which represent whether the failure times happened before or after the observation point. In this work, the observed bivariate data can be written with the following form {x, £_1=I(X1≤ x), £_2=I(X2≤ x)}.The corresponding optimal design problems for parameter estimation under this type of bivariate data are discussed. For this kind of the multivariate responses with explanatory variables, their marginal distributions may be from different distributions. Copula model is a way to formulate the relationship of these responses, and the association between pairs of responses. Copula models for bivariate binary data are considered useful in practice due to its flexibility. In this dissertation for bivariate binary data, the marginal functions are assumed from exponential or Weibull distributions and two assumptions, independent or correlated, about the joint function between variables are considered. When the bivariate binary data is assumed correlated, the Clayton copula model is used as the joint cumulative distribution function. There are few works addressed the optimal design problems for bivariate binary data with copula models. The D-optimal designs aim at minimizing the volume of the confidence ellipsoid for estimating unknown parameters including the association parameter in bivariate copula models. They are used to determine the best observation points. Moreover, the Ds-optimal designs are mainly used for estimation of the important association parameter in Clayton model. The D- and Ds-optimal designs for the above copula model are found through the general equivalence theorem with numerical algorithm. Under different model assumptions, it is observed that the number of support points for D-optimal designs is at most as the number of model parameters for the numerical results. When the difference between the marginal distributions and the association are significant, the association becomes an influential factor which makes the number of supports gets larger. The performances of estimation based on optimal designs are reasonably well by simulation studies. In survival experiments, the experimenter customarily takes trials at some specific points such as the position of the 25, 50 and 75 percentile of distributions. Hence, we consider the design efficiencies when the design points for trials are at three or four particular percentiles. Although it is common in practice to take trials at several quantile positions, the allocations of the proportion of sample size also have great influence on the experimental results. To use a locally optimal design in practice, the prior information for models or parameters are needed. In case there is not enough prior knowledge about the models or parameters, it would be more flexible to use sequential experiments to obtain information in several stages. Hence with robustness consideration, a sequential procedure is proposed by combining D- and Ds-optimal designs under independent or correlated distribution in different stages of the experiment. The simulation results based on the sequential procedure are compared with those by the one step procedures. When the optimal designs obtained from an incorrect prior parameter values or distributions, those results may have poor efficiencies. The sample mean of estimators and corresponding optimal designs obtained from sequential procedure are close to the true values and the corresponding efficiencies are close to 1. Huster (1989) analyzed the corresponding modeling problems for the paired survival data and applied to the Diabetic Retinopathy Study. Huster (1989) considered the exponential and Weibull distributions as possible marginal distributions and the Clayton model as the joint function for the Diabetic Retinopathy data. This data was conducted by the National Eye Institute to assess the effectiveness of laser photocoagulation in delaying the onset of blindness in patients with diabetic retinopathy. This study can be viewed as a prior experiment and provide the experimenter some useful guidelines for collecting data in future studies. As an application with Diabetic Retinopathy Study, we develop optimal designs to collect suitable data and information for estimating the unknown model parameters. In the second part of this work, the optimal design problems for parameter estimations are considered for the type of proportional data. The nonlinear model, based on Jorgensen (1997) and named the dispersion model, provides a flexible class of non-normal distributions and is considered in this research. It can be applied in binary or count responses, as well as proportional outcomes. For continuous proportional data where responses are confined within the interval (0,1), the simplex dispersion model is considered here. D-optimal designs obtained through the corresponding equivalence theorem and the numerical results are presented. In the development of classical optimal design theory, weighted polynomial regression models with variance functions which depend on the explanatory variable have played an important role. The problem of constructing locally D-optimal designs for simplex dispersion model can be viewed as a weighted polynomial regression model with specific variance function. Due to the complex form of the weight function in the information matrix is considered as a rational function, an approximation of the weight function and the corresponding optimal designs are obtained with different parameters. These optimal designs are compared with those using the original weight function.

This paper is concerned with the problem of computing theapproximate D-optimal design for polynomial regression with weight function w(x)>0 on the design interval I=[m_0-a,m_0+a]. It is shown that if w'(x)/w(x) is a rational function on I and a is close to zero, then the problem of constructing D-optimal designs can be transformed into a differential equation problem leading us to a certain matrix including a finite number of auxiliary unknown constants, which can be approximated by a Taylor expansion. We provide a recursive algorithm to compute Taylor expansion of these constants. Moreover, the D-optimal interior support points are the zeros of a polynomial which has coefficients that can be computed from a linear system.

This paper is concerned with the problem of constructing A-optimal design for polynomial regression with analytic weight function on the interval [m-a,m+a]. It is shown that the structure of the optimal design depends on a and weight function only, as a close to 0. Moreover, if the weight function is an analytic function a, then a scaled version of optimal support points and weights is analytic functions of a at $a=0$. We make use of a Taylor expansion which coefficients can be determined recursively, for calculating the A-optimal designs.