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Functionally Graded Materials
Functionally graded materials (FGMs) are inhomogeneous materials, consisting of two (or more) different materials, engineered to have a continuously varying spatial composition profile. The focus of our research has been on the development of a robust methodology for the optimization of material distribution of two-phase FGMs. The performance of a two-phase functionally graded component that is subjected to various boundary conditions and thermomechanical loads is maximized by optimizing the volume fraction profile of one of its material phases. We have investigated the optimization of metal/ceramic FGM components for high temperature applications and metal/metal FGMs for high heat flux applications. We have also studied the tuning of natural frequencies of FGMs.
In the proposed methodology, all candidate designs are evaluated using a mesh-free method, namely the Element-free Galerkin (EFG) method. The spatial distribution of constituent volume fraction, which is to be tailored for optimal performance, is obtained by a range-restricted piecewise bicubic interpolation of volume fractions defined at a finite number of grid points, as demonstrated in Figure 1.
Figure 1: Determination of the spatial volume fraction profile from nodal volume fraction values using range-restricted bicubic interpolation.
The effective material properties at a point in the domain are estimated from the local volume fractions of the material constituents using the Mori-Tanaka, self-consistent and Hashin-Shtrikman homogenization schemes. The volume fraction distribution is optimized using a real-coded genetic algorithm (GA) which acts on the volume fractions values located at the spatial locations defined by the nodes of the aforementioned grid. The GA fitness and constraint information is constructed from the EFG analysis. Both single and multi-objective problems are tackled using standard GA routines.
Prior to utilizing the proposed approach for FGM optimization, the EFG method is validated using known exact solutions for functionally graded plates. A sample result from the EFG free vibration validation study is shown in Figure 2.
Figure 2: (a) Mode shape, (b) displacement u2 , (c) longitudinal stress s11 and (d) shear stress s12 corresponding to the sixth natural frequency of a tungsten/copper FGM plate.
The figure displays comparisons of the exact and EFG solutions for the through-thickness trends in the field variables for a tungsten/copper simply supported FGM. The results shown, which correspond to the sixth natural frequency of the plate, are computed with 120 EFG analysis nodes and compare very well with the exact solution. To ensure the accuracy of the EFG method as applied to the analysis of thermally loaded FGM components, the numerical results obtained by the EFG method are compared to the exact solution for a steeply graded aluminum/silicon carbide FGM plate subjected to high temperatures. A sample result from the study, shown in Figure 3, displays contour plots of the effective, or von Mises, stress obtained by each method. Although only 216 nodes are used in the EFG analysis, the results obtained are nearly indistinguishable from those given by the exact solution.
Figure 3: Contour plots of the effective stress for a steeply graded aluminum/silicon carbide FGM plate obtained by (a) the EFG method and (b) the exact solution.
With confidence established in the EFG method for accurately simulating the response of FGM components, the proposed methodology is put to the test via numerous example problems. One of the example problems performed is the unconstrained minimization of the peak effective stress of a nickel/alumina FGM that is uniformly cooled from a high fabrication temperature. Through the application of the proposed methodology, the grading architecture was optimized to yield significant reductions in the peak effective stress of the component. The obtained volume fraction profile, as well as the spatial variation of the effective stress and GA trends, is given in Figure 4.
Figure 4: (a) Fitness of best individual and average fitness of population as a function of generation number for a sample run of the GA, (b) surface plot of the optimized alumina volume fraction, (c) density plot of optimized ceramic volume fraction and corresponding (d) effective stress field for an optimized nickel/alumina FGM component.
For the design shown in the figure, the peak effective stress is 129.2 MPa, a significant improvement over results obtained via other methodologies currently employed in scientific literature. The design obtained also yields a 55.9% reduction in peak effective stress as compared to a simple linear grading of the material constituents through the thickness of the component in the design region.
Another single objective optimization tackled with the proposed methodology, albeit with constraints, is the mass minimization of a clamped-simply supported tungsten/copper beam. The mass of the FGM beam is to be minimized whilst ensuring that the natural frequencies of the beam lie outside two frequency constraint zones, the first between 5000 Hz and 8000 Hz, and the other being from 13000 Hz to 16000 Hz. The results of the optimization are shown in Figure
Figure 5: Mass minimization of a tungsten/copper FGM beam with constraints on the natural frequencies, (a) fitness trends for a sample run of the GA, (b) density plot of the optimized tungsten volume fraction distribution and (c) frequency response function of the optimized beam when subjected to base excitations.
With the volume fraction profile shown in the figure, the beam satisfies all of the frequency constraints and contains 50.9% tungsten by mass, tungsten being the heavier of the two material constituents. Both monolithic copper and tungsten beams violate the frequency constraints, making these options infeasible designs. As can be seen in Figure 5, the volume fraction profile, while symmetric, is rather complex and would be hard to arrive at without a stout FGM optimization procedure.
The proposed methodology can also be extended to perform multi-objective optimization of FGMs with temperature-dependent material properties undergoing thermomechanical processes. The movie below in Figure 6 displays the result of simultaneously minimizing the mass and maximizing the factor of safety of a tungsten/copper alloy FGM subjected to a localized intense heat flux applied to the first 0.01m of the top edge of the component.
Figure 6: Density plots of the volume fraction profile for Pareto-optimal solutions obtained for the simultaneous mass minimization and factor of safety maximization of a tungsten/copper alloy FGM subjected to an intense heat flux [Click here to download Windows Media Video].
The spatial grading profiles obtained for the Pareto-optimal, or optimal trade-off designs, are shown as the movie progresses. The obtained Pareto-optimal designs all outperform obvious material distribution profiles, and more importantly, supply designers with a number of excellent designs to choose from.
Through the application of the developed FGM optimization technique to numerous example problems, we have demonstrated the merits of our methodology in designing superior performing FGM components. The combination of range-restricted bicubic interpolation of the volume fraction field, EFG analysis and optimization via genetic algorithms has proven to be most effective. Not only is the methodology robust, it is extremely flexible and is easily extended to the optimization of FGMs for applications outside of the realms of steady-state vibration and thermoelasticty considered in our works. |
