79,051 research outputs found
On the numerical solution of a time-dependent shape optimization problem for the heat equation
This article is concerned with the solution of a time-dependent shape identification problem. Specifically we consider the heat equation in a domain, which contains a time-dependent inclusion of zero temperature. The objective is to detect this inclusion from the given temperature and heat flux at the exterior boundary of the domain. To this end, for a given temperature at the exterior boundary, the mismatch of the Neumann data is minimized. This time-dependent shape optimization problem is then solved by a gradient-based optimization method. Numerical results are presented which validate 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
A Method for Geometry Optimization in a Simple Model of Two-Dimensional Heat Transfer
This investigation is motivated by the problem of optimal design of cooling
elements in modern battery systems. We consider a simple model of
two-dimensional steady-state heat conduction described by elliptic partial
differential equations and involving a one-dimensional cooling element
represented by a contour on which interface boundary conditions are specified.
The problem consists in finding an optimal shape of the cooling element which
will ensure that the solution in a given region is close (in the least squares
sense) to some prescribed target distribution. We formulate this problem as
PDE-constrained optimization and the locally optimal contour shapes are found
using a gradient-based descent algorithm in which the Sobolev shape gradients
are obtained using methods of the shape-differential calculus. The main novelty
of this work is an accurate and efficient approach to the evaluation of the
shape gradients based on a boundary-integral formulation which exploits certain
analytical properties of the solution and does not require grids adapted to the
contour. This approach is thoroughly validated and optimization results
obtained in different test problems exhibit nontrivial shapes of the computed
optimal contours.Comment: Accepted for publication in "SIAM Journal on Scientific Computing"
(31 pages, 9 figures
What is the optimal shape of a fin for one dimensional heat conduction?
This article is concerned with the shape of small devices used to control the
heat flowing between a solid and a fluid phase, usually called \textsl{fin}.
The temperature along a fin in stationary regime is modeled by a
one-dimensional Sturm-Liouville equation whose coefficients strongly depend on
its geometrical features. We are interested in the following issue: is there
any optimal shape maximizing the heat flux at the inlet of the fin? Two
relevant constraints are examined, by imposing either its volume or its
surface, and analytical nonexistence results are proved for both problems.
Furthermore, using specific perturbations, we explicitly compute the optimal
values and construct maximizing sequences. We show in particular that the
optimal heat flux at the inlet is infinite in the first case and finite in the
second one. Finally, we provide several extensions of these results for more
general models of heat conduction, as well as several numerical illustrations
Inverse Design Based on Nonlinear Thermoelastic Material Models Applied to Injection Molding
This paper describes an inverse shape design method for thermoelastic bodies.
With a known equilibrium shape as input, the focus of this paper is the
determination of the corresponding initial shape of a body undergoing thermal
expansion or contraction, as well as nonlinear elastic deformations. A
distinguishing feature of the described method lies in its capability to
approximately prescribe an initial heterogeneous temperature distribution as
well as an initial stress field even though the initial shape is unknown. At
the core of the method, there is a system of nonlinear partial differential
equations. They are discretized and solved with the finite element method or
isogeometric analysis. In order to better integrate the method with
application-oriented simulations, an iterative procedure is described that
allows fine-tuning of the results. The method was motivated by an inverse
cavity design problem in injection molding applications. Its use in this field
is specifically highlighted, but the general description is kept independent of
the application to simplify its adaptation to a wider range of use cases.Comment: 22 pages, 13 figure
A "poor man's" approach for high-resolution three-dimensional topology optimization of natural convection problems
This paper treats topology optimization of natural convection problems. A
simplified model is suggested to describe the flow of an incompressible fluid
in steady state conditions, similar to Darcy's law for fluid flow in porous
media. The equations for the fluid flow are coupled to the thermal
convection-diffusion equation through the Boussinesq approximation. The coupled
non-linear system of equations is discretized with stabilized finite elements
and solved in a parallel framework that allows for the optimization of high
resolution three-dimensional problems. A density-based topology optimization
approach is used, where a two-material interpolation scheme is applied to both
the permeability and conductivity of the distributed material. Due to the
simplified model, the proposed methodology allows for a significant reduction
of the computational effort required in the optimization. At the same time, it
is significantly more accurate than even simpler models that rely on convection
boundary conditions based on Newton's law of cooling. The methodology discussed
herein is applied to the optimization-based design of three-dimensional heat
sinks. The final designs are formally compared with results of previous work
obtained from solving the full set of Navier-Stokes equations. The results are
compared in terms of performance of the optimized designs and computational
cost. The computational time is shown to be decreased to around 5-20% in terms
of core-hours, allowing for the possibility of generating an optimized design
during the workday on a small computational cluster and overnight on a high-end
desktop
Two optimization problems in thermal insulation
We consider two optimization problems in thermal insulation: in both cases
the goal is to find a thin layer around the boundary of the thermal body which
gives the best insulation. The total mass of the insulating material is
prescribed.. The first problem deals with the case in which a given heat source
is present, while in the second one there are no heat sources and the goal is
to have the slowest decay of the temperature. In both cases an optimal
distribution of the insulator around the thermal body exists; when the body has
a circular symmetry, in the first case a constant heat source gives a constant
thickness as the optimal solution, while surprisingly this is not the case in
the second problem, where the circular symmetry of the optimal insulating layer
depends on the total quantity of insulator at our disposal. A symmetry breaking
occurs when this total quantity is below a certain threshold. Some numerical
computations are also provided, together with a list of open questions.Comment: 11 pages, 7 figures, published article on Notices Amer. Math. Soc. is
available at
http://www.ams.org/publications/journals/notices/201708/rnoti-p830.pd
- …