Optimization of current carrying multicables

Intense electric currents in cable bundles contribute to hotspot generation and overheating of essential car elements, especially in connecting structures. An important aspect in this context is the influence of the positioning of wires in cable harnesses. In order to find an appropriate multicable layout with minimized maximum temperatures, we formulate an optimization problem. Depending on the packing density of the cable bundle, it is solved via different optimization strategies: in case of loosely packed cable bundles solely by a gradient based strategy (shape optimization), densely packed ones by arrangement heuristics combined with a standard genetic algorithm, others by mixed strategies. In the simulation model, the temperature dependence of electric resistances and different parameter values for the multitude of subdomains are respected. Convective and radiative effects are summarized by a heat transfer coefficient in a nonlinear boundary condition. Finite elements in combination with an interior-point method and a genetic algorithm allow the solution of the optimization problem for a large number of cable bundle types. Furthermore, we present an adjoint method for the solution of the shape optimization problem. The jumps at the interfaces of different materials are essential for the Hadamard representation of the shape gradient. Numerical experiments are carried out to demonstrate the feasibility and scope of the present approach

Optimization of current carrying multicables

High currents in cable bundles contribute to hotspot generation and overheating of essential car elements, especially in connecting structures. An important aspect in this context is the influence of the positioning of wires in cable harnesses. In order to find an appropriate multicable layout with minimized maximum temperatures, we formulate a corresponding optimization problem. Depending on the packing density of the cable bundle, it is solved via different optimization strategies: in case of loosely packed cable bundles solely by a gradient-based strategy (shape optimization), densely packed ones by arrangement heuristics combined with a standard genetic algorithm, others by mixed strategies. In the simulation model, temperature dependence of the electric conductor resistances and different parameter values for the multitude of subdomains are respected in the governing semilinear and piecewise defined equation. Convective and radiative effects are summarized by a heat transfer coefficient in a nonlinear boundary condition at the exterior multicable surface. Finite elements in combination with an interior-point method and a genetic algorithm allow the solution of the optimization problem for a large number of cable bundle types. Furthermore, we present an adjoint method for the solution of the shape optimization problem. The jumps at the interfaces of different materials are essential for the Hadamard representation of the shape gradient. Numerical experiments are carried out to demonstrate the feasibility and scope of the present approach

Optimization of modular wiring harnesses by means of regression models for temperature prediction of wire bundles

Automotive wiring harnesses have become heavier and more complex due to their increasing number of electrical components. It is now desired to reduce their mass of copper. For this purpose, experimentation can be partially replaced by simulation, but it is still impossible to exhaustively simulate all of the combinations of modular wiring harness. This proposed approach consists of carrying out simulations using the FEM method and using their results to create regression models. Polynomial formulae can give the same information as simulations within a clearly reduced time and satisfying accuracy. An optimization algorithm introduced in this study will use them to assign new cable cross-sections of harnesses considering their currents and the ambient temperature.Postprint (author's final draft

Tractability of the Quasi-Monte Carlo quadrature with Halton points for elliptic PDEs with random diffusion

This article is dedicated to the computation of the moments of the solution to stochastic partial differential equations with log-normal distributed diffusion coefficient by the Quasi-Monte Carlo method. Our main result is the polynomial tractability for the Quasi-Monte Carlo method based on the Halton sequence. As a by-product, we obtain also the strong tractability of stochastic partial differential equations with uniformly elliptic diffusion coefficient by the Quasi-Monte Carlo method. Numerical experiments are given to validate the theoretical findings

Stabilization of the trial method for the Bernoulli problem in case of prescribed Dirichlet data

We apply the trial method for the solution of Bernoulli's free boundary problem when the Dirichlet boundary condition is imposed for the solution of the underlying Laplace equation and the free boundary is updated according to the Neumann boundary condition. The Dirichlet boundary value problem for the Laplacian is solved by an exponentially convergent boundary element method. The update rule for the free boundary is derived from the linearization of the Neumann data around the actual free boundary. With the help of shape sensitivity analysis and Banach’s fixed-point theorem, we shed light on the convergence of the respective trial method. Especially, we derive a stabilized version of this trial method. Numerical examples validate the theoretical findings

Multilevel accelerated quadrature for PDEs with log-normal distributed random coefficient

This article is dedicated to multilevel quadrature methods for the rapid solution of stochastic partial differential equations with a log-normal distributed diffusion coefficient. The key idea of these approaches is a sparse grid approximation of the occurring product space between the stochastic and the spatial variable. We develop the mathematical theory and present error estimates for the computation of the solution's statistical moments with focus on the mean and variance. Especially, the present framework covers the multilevel Monte Carlo method and the multilevel quasi Monte Carlo method as special cases. The theoretical findings are supplemented by numerical experiments

The H2-wavelet method

the present paper, we introduce the H2-wavelet method for the fast solution of nonlocal operator equations on unstructured meshes. On the given mesh, we construct a wavelet basis which provides vanishing moments with respect to the traces of polynomials in the space. With this basis at hand, the system matrix in wavelet coordinates is compressed to O(N log N) relevant matrix coefficients, where N denotes the number of boundary elements. The compressed system matrix is computed with nearly linear complexity by using the fast H2-matrix approach. Numerical results in three spatial dimensions validate that we succeeded in developing a fast wavelet Galerkin scheme on unstructured triangular or quadrangular meshes

The H2-wavelet method

the present paper, we introduce the H2-wavelet method for the fast solution of nonlocal operator equations on unstructured meshes. On the given mesh, we construct a wavelet basis which provides vanishing moments with respect to the traces of polynomials in the space. With this basis at hand, the system matrix in wavelet coordinates is compressed to O(N log N) relevant matrix coefficients, where N denotes the number of boundary elements. The compressed system matrix is computed with nearly linear complexity by using the fast H2-matrix approach. Numerical results in three spatial dimensions validate that we succeeded in developing a fast wavelet Galerkin scheme on unstructured triangular or quadrangular meshes

A novel optimization methodology of modular wiring harnesses in modern vehicles : weight reduction and safe operation

The weight of electric and electronic components of cars has been uninterruptedly increasing through the last decades, and thus the weight of their wiring harnesses. This fact has awakened the interest of car manufacturers on the weight and cost optimization of automotive wiring harnesses . For this reason, this dissertation discusses and develops approaches to reduce the amount of copper for the purpose of current conduction, i.e. the cross-sections of all of the wires of the car, without endangering safety. On the one hand, harnesses must withstand continuous operation currents. On account of this, it is necessary to know the characteristic flow of current of the in-vehicle electrical network. Nevertheless. the huge quantity of available combinations of equipment of the car produces a proportional variety of customer-specific wiring hamesses, and makes it unfeasible to simulate all of them. This thesis points attention on specific segments of the wiring harnesses. Sorne of them can have many possible compositions, which are related to the customer's car settings. Since computation time is a limiting factor here, it is proposed to predict the bundle heating behaviors by means of response surfaces, obtained from a set of finite element simulation results and the least squares method. On the other hand, the correct wire sizes must ensure that they are protected by their associated melting fuses, so that their maximum acceptable temperature is not exceeded after short circuits. Since many wires in cars are connected to other wires with splices, or may suffer short-circuits in their electric loads, these short-circuits can flow across different wires. In modular wiring harnesses, each of the wires can have different lengths and different installation ratios, their cross-section affects the cost of the wire harness with different importance, as well as the short circuit and the final temperature of the wire. The finite volume method is used to simulate the short circuit of series-connected wires. Finally, non-linear optimization is used to find the mínimum cross­ sections of wires respecting the constraints of maximum temperature and mínimum short-circuit current. Finally, these two different criteria for optima! wire dimensioning are combined in the analysis of the on-board network of the vehicle in order to make a complete weight and cost minimization of the cable harnesses in a particular vehicle, considering also its modularity of loads.El pes dels components elèctrics i electrònics deis automòbils ha crescut ininterrompudament al llarg de les darreres dècades, i conseqüentment ho han fet també els seus feixos de cables. Aquest fet ha despertat entre els fabricants de turismes un elevat interès en la minimització del pes i dels costos del cablejat del vehicle. Per aquest motiu, aquesta tesi desenvolupa mètodes per reduir la quantitat de coure destinat a la conducció de corrent, és a dir, les seccions de tots els fils elèctrics dins el cotxe, sense posar en risc la seguretat. Per una banda, els feixos han de resistir els corrents d'operació continuada. Per a aquest propòsit, cal conèixer el flux de corrents característic de la xarxa de bord del vehicle. No obstant, la immensa quantitat de combinacions de diferents equipaments del vehicle produeix proporcionalment una enorme varietat de feixos personalitzats per als clients, fet que fa inviable simular totes aquestes combinacions . El primer dels mètodes d'optimització que es proposen en aquesta tesi estudia segments dels feixos de cables per separat un a un. Alguns d'ells poden tenir diferents composicions de fils en funció de la configuració aplicada pel client. Com que el temps de calcul és un factor limitant, es proposa predir el comportament tèrmic dels segments per mitja de superfícies resposta, que s'obtenen a través del mètode deis mínims quadrats i un conjunt de resultats de simulació de feixos pel mètode dels elements finits. Per altra banda, les correctes seccions dels fils han de ser tals que els curtcircuits i les sobrecarregues no puguin malmetre'ls, gracies a la correcta coordinació amb els fusibles destinats a protegir-los. Atès que molts fils estan connectats amb altres fils per mitja d'unions soldades i que molts curtcircuits són provocats directament en bornes de les carregues elèctriques, els curtcircuits poden fluir a través de fils diferenciats connectats en serie. Als feixos modulars, cadascun deis fils té diferents longituds i ratis d'instal·lació. És per aquest darrer motiu que llur secció afecta de diferent manera al cost total del conjunt deis feixos de cables deis cotxes venuts . De la mateixa manera, les seves longituds diferents fan que les variacions en les seccions alterin els curtcircuits resultants amb diferent sensibilitat. És per això que es fa servir optimització no lineal per trobar les seccions separades de cadascun dels fils connectats en serie a través dels quals poden passar curtcircuits. Per a aquesta fi es fan simulacions en volums finits i models energètics dels fusibles integrades dins de l'optimització no lineal. Finalment, aquestes dues vies de dimensionament es combinen dins una anàlisi íntegra de la xarxa de bord per dimensionar de forma òptima cadascun dels fils del vehicle, tenint en compte les interconnexions entre feixos i totes les combinacions d'equipament