Multi-dimensional optimization of functionally graded material composition using polynomial expansion of the volume fraction

Significance Statement

A functionally graded material is a composite material with two or more materials with varying volume fractions through the dimensions of a structure. The continuous variation of the constituent materials overcomes the problems of matrix cracking, delamination, and stress concentration that are common in multilayered composites reference to the difference in mechanical and thermal characteristics of the constituent layers.

The idea of functionally graded material was incepted with an aim of protecting space structured from excessive temperatures that spacecraft experience on reentry to the Earth’s atmosphere. Ceramic tiles, which have been used as insulators, they suffer from de-bonding and cracking at the interface between the structure and the tiles reference to difference in the thermal expansion coefficients between the structure and the tiles. Therefore, there has been a need to have a material with gradual variation of the thermal attributes in order to sustain the excess heat on the high temperature side.

Based on the research works relating to the field of functionally graded material tailoring, it is evident that optimization methods have been based on exponential or power laws, and on the grid approach. The grid approach is computationally inefficient, while the power law puts a limit on the volume fraction to follow a particular monotonic variation throughout the structure.

Therefore, Omar Hussein and Prof.Sameer B. Mulani at the University of Alabama proposed an efficient optimization method to be applied in the design of functionally graded structures. The method was to be considered as an intermediate approach between the two methods and would not have constraints on the behavior of the volume fraction variation over the structure. Their work is published in Structural and Multidisciplinary Optimization.

Functionally graded materials are designed with two or more materials mixed together in order to obtain a composite material with spatially varying thermal and mechanical properties. The attributes of the resulting materials are controlled by the volume fraction of each constituent element. To spatially tailor the material attributes, the volume fraction should be tailored.

The authors adopted a method based on applying polynomials expansion of the volume fraction of the constituent elements. In the method, the design variables were the coefficients of the polynomial expansion, which were established through an optimization process.

The approach allowed for a more flexible design compared to power or the exponential law and this led to a more optimal design. It also demands less design variables as opposed to the grid based method.

The proposed method was applied to a simply supported aluminum plate reinforced by silicon carbide nanoparticles. Considering that the cost of the nano-reinforcement was too high, the aim of the approach was to minimize the overall volume fraction of the reinforcement by optimizing its distribution.

The results of the study indicated that using material grading could save between 20-80% of the reinforcement required counting on the nature of the problem. Reinforcing aluminum-based structure with silicon carbide could enhance its mechanical performance without additional weight. This appears to be very promising for aerospace applications.

functionally graded material composition polynomial expansion volume fraction- Advances in Engineering

About the author

Mr.Omar is a Ph.D. Student in the Aerospace engineering and Mechanics Department at the University of Alabama, Tuscaloosa, USA. He received his M.Sc degree in 2013 from Cairo University, Egypt. His research interests lie in the fields of structural analysis, nanocomposites, optimization, and aeroelasticity.

About the author

Dr. Mulani is an assistant professor of the Aerospace engineering and Mechanics Department at the University of Alabama, Tuscaloosa, USA.  His research interests include aerospace structures, finite element methods, fluid-structure interaction, numerical methods, optimization, random vibrations, uncertainty quantification and vibro-acoustics.


Omar S. Hussein and Sameer B. Mulani. Multi-dimensional optimization of functionally graded material composition using polynomial expansion of the volume fraction. Structural and Multidisciplinary Optimization, volume 56 (2017), pages 271–284.

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