The paper considers various structures designed to shield temperature sensitive devices from the heat generated by a given distributed source. These structures should redistribute the heat and control the total heat dissipation, in order to maintain a prescribed temperature profile in a certain region of the domain. We assume there are several materials available, each with different constants of heat conductivity, and that we are allowed to mix them arbitrarily. It is known that optimal structures consist of laminate composites that are allowed to vary within the design domain. The paper discusses optimal distributions of these composites for different settings of the problem that include: prescribed volume fractions and various types of boundary conditions. We solve the problem numerically using the method of finite elements.
Structural Optimization, Finite Elements, Laminate Composites, Optimal Control
The design of the experiments presented in this paper is considered to be a tool used to investigate different material distributions based on a variety of loadings, boundary conditions, domain shapes, etc. The results that are presented in this paper are based on the model for thermal conductivity. However, the techniques used to generate these results are not confined exclusively to this model.
The results of this investigation show the importance of laminate material in optimal structure. We find that with the introduction of laminate materials in the design, we typically have more than 10-X improvement in our goal functional over a homogeneous, single material distribution. It is the way in which these laminate material appear in optimal structures that the paper investigates.
Figure 1.a: Interior goal function with presecribed boundary conditions. | Figure 1.b: Boundary goal function with presecribed boundary conditions. |
Let us start with two example problems. The first example is depicted in the Figure 1.a. The question is, how can we distribute two different materials in a domain O, so that the integral over the sub-domain O^{*} of the squared difference between the goal function and the potential u(x) is minimized,
In (1), u(x) is the solution of the two-dimensional conductivity equation and r(x) is the weight function
For this problem, we assume that the boundary conditions are fixed, and we are free to distribute the materials inside O.
In general, a homogeneous isotropic material distribution can not minimize this functional. The expected result is a structure that consists of both materials, and contains regions of laminates. These anisotropic regions are needed in order to direct the internal currents against the imposed temperature gradient or external loadings.
In the second problem, depicted in Figure 1.b, boundary conditions are imposed on O and the total volume fractions of the materials are prescribed. The considered problem is the minimization of the squared difference between the temperature on a segment of the boundary, dO_{S},
Since the potential u in O is governed by the steady state conductivity equation, once the boundary conditions are fixed, u is determined. In these two example problems, we wish to control the solution of the conductivity equation, not through variation of the boundary conditions or loadings but through the distribution of the materials within the domain. In this manner, we hope to achieve the desired goal, that would otherwise not be obtainable with the prescribed boundary conditions.
For the examples presented in this paper, the optimization problem can be stated as the minimization of the integral over O of the continuous function F = F(x,u) and the integral around the boundary of O of the continuous function P = P(u,u_{n}),
where x is a point in the domain O. The potential u in (4) must also satisfy the two-dimensional dimensional conductivity equation
In (5), K(x) is the 2x2 conductivity tensor for the material distribution, g_{1} describes Dirichlet boundary conditions and g_{2} describes Neumann boundary conditions. To find the approximation to the solution of (5), the Finite Element Method (FEM) is used to find the approximate solution of the equivalent weak form;
The finite element solution provides a piece-wise linear approximation to the solution of the weak form of the conductivity equation. The FEM solution to (6) is computed using the standard linear system solver [4], optimized for sparse systems. <\P>
To solve the minimization problem (4) subject to the differential constraint (5), we introduce the Lagrange multiplier L(x). With the Lagrange multiplier, we can reformulate the constrained minimization problem (4) as:
Applying standard variational techniques to (7), i.e., the stationary requirement that d_{L} J = 0, where d_{L} J is the variation of J resulting from a variation of u, u, we arrive at the following boundary value problem for L(x),
To compute L(x), we used the same finite element solver used to solve the boundary value problem for u. Figure 3 shows the numerical solution of nabla u and nabla L for example 1 described below (see section (6)).
At each point (x) in O, we must determine the material properties tensor K(x) in order to minimize the functional in (7). We assume that laminate structures will appear in an optimal design, together with regions of pure material [2]. The laminate structures are represented by an anisotropic conductivity tensor, K(x). The eigenvectors of K correspond to the normal and tangential directions of the laminates and the eigenvalues of K correspond to the arithmetic and harmonic mean of the two materials,
In (9), m_{1} is the relative volume fraction of the first material, 1 - m_{1} is the relative volume fraction of the second material, k_{1} is the scalar conductivity of the first material and k_{2} is the scalar conductivity of the second material (k_{1} < k_{2}). For these problems, we treat the orientation of the eigenvectors and the volume fraction of the materials as the controls. In addition, we assume that K(x) is varying continuously from point to point in O.
To specify the laminate structure, we must compute the relative volume fraction of the first material m_{1} and varphi, the orientation of the structure (refer to figure 2). By applying homogenization techniques to the functional J, we determined the following condition for the volume fraction of the first material m_{1};
In (10), the angle t = t(x) is the angle between the vectors nabla u and nabla L. One can also show that the direction of the lamination, varphi = varphi(x), bisects the angle between nabla u and nabla L [2] (see figure 2).
Figure 2: K(x) = K(x,m_{1},varphi) |
Therefore, to minimize J, we assume a homogeneous distribution of one of the materials in O and solve the boundary value problems for u and L. From these solutions, we numerically compute nabla u and nabla L and compute a new conductivity tensor K(x). This procedure is then repeated for the updated K tensor. The iterations then continue until the difference in the functional J between iterations is less than a specified tolerance.
To demonstrate the effectiveness of this method, we examine two a numerical examples. In the first example, we are given two isotropic conducting materials, with scalar conductivities k_{1} = 1 and k_{2} = 10, and do not place any restrictions on the volume fraction for either material. The domain O (0 <= x <= 1 and 0 <= y <= 1) is loaded with the localized heat source
and the potential u is set to zero on the boundary, u|_{ðO} = 0. The goal for this example is to minimize the potential located at the heat source and redistribute the heat to the point located at the opposite side of O, i.e., we want to minimize the following functional,
where the weight function r(x,y) is defined as
The gradients for u and L for the initial material distribution are displayed in the upper-left in figure 3. In the lower-left in figure 3, the direction of the minimum eigenvalue of K(x), along with nabla u and nabla L, is displayed. In the lower-right in figure 3, the volume fraction of the first material is shown. In this picture, blue represents regions of only pure material 1 (m_{1} = 1), red represents pure material 2 (m_{1} = 0) and the color gradient represents regions where a mixture of the two materials occur (0 < m_{1} 1).
Comparing the two pictures illustrated in the lower portion of figure 3, we see how the materials are distributed in order to achieve the goal. The region of pure material 1, the blue region, acts to insulate the localized source from the boundary. The region of pure material 2, the red region, act as a conduction path between the source and the point in O to be heated. In between these two regions, the laminations direct the flux from the source and toward the desired location in the domain. On the lower-left in figure 3, the black line represents the direction of minimum eigenvalue of K.
Figure 3: Material Distribution for Numerical Example 1 upper left -- nabla u(x) and nabla L, lower left -- lamination direction, |
upper right -- evaluation of J lower right -- Volume fraction m_{1} |
The graph of the functional J in (12) for the first 10 iterations is shown in the upper-right in figure 3. In four iterations, we see that the functional has obtained a minimum, and further iterations do not improve the design. In figure 4, the left picture illustrates the potential u in O for a homogeneous distribution of the first material and the right picture illustrates the potential u in O after the fourth iteration when the minimum has been obtained.
Figure 4: u(x) for example problem 1. left -- iteration 0, |
right -- iteration 4 |
In the second example, we are given two isotropic conducting materials, with scalar conductivities k_{1} = 1 and k_{2} = 10, and do not place any restrictions on the volume fraction for either material. The domain O (0 <= x <= 1 and 0 <= y <= 1) is loaded with the localized heat source
and the potential u is set to zero on the boundary, u|_{pO} = 0. The goal for this example is reduce the peak load in O by distributing the load to a small area surrounding the center of the source. The goal functional for this example is,
Figure 5: Numerical Example 2 left -- lamination direction |
right -- evaluation of J |
Figure 6: m_{1} for Numerical Example 2 upper left -- iteration 0, lower left -- iteration 6, |
upper right -- iteration 4 lower right -- iteration 9 |
On the left in figure 5, we see that laminations form in the region surrounding the localized source and tend to direct to the heat flux to stay within a neighborhood of the source. The evolution of material distribution, for the first 10 iterations, is illustrated in figure 6. Here we see that the source is insulated from the boundary by the pure material with the lower thermal conductivity and the best conductor is located at the center of the source. The laminations are present in the region separating these two regions, and act to direct the flux in the area surrounding the source.
The graph of the functional J in (15) for the first 10 iterations is shown on the right in figure 5. In ten iterations, we see that the functional has obtained a minimum, and further iterations fall to improve the design. In figure 7, the left picture illustrates the potential u in O for a homogeneous distribution of the first material and the right picture illustrates the potential u in O after the tenth iteration when the minimum is obtained. Figure 7 illustrates how the material distribution acts to lower the maximum potential in the domain and form a more uniform temperature gradient near the localized heat source.
Figure 7: u(x) for example problem 2 left -- iteration 1, |
right -- iteration 9 |
In addition to solving the problem for various material distributions and boundary conditions, the desired field functional can be used to mimic some ideal physical situation. For example, in terms of thermal conductivity, this might correspond to directing thermal current away from possible temperature sensitive devices toward heat dissipating devices, i.e., heat sinks. Alternatively, a numerical scheme can be developed to find a sub-optimal solution. For example, the sub-optimal structure may have piece-wise constant properties, as opposed to the continuously varying properties of the truly optimal structure. Since a structure with continuously varying properties may not be realizable, the sub-optimal structure may be the more desirable solution.
For the problem of minimizing the total energy stored in a domain with limited resources: we are given a small volume fraction of a good conductor and asked how is this material distributed in order to maximize its contribution to the energy minimization? In addition to varying the volume fractions of the materials, different boundary conditions are imposed on the structures. This variation should also provide insight into how the materials orient themselves in order to direct the fields within O, from sub-domains that are less desirable to areas of greater importance. These are only a few of the directions that this type of research tool can be used to examine how materials are distributed in optimal structures.