Thermo-elastic optimization of material distribution of functionally graded structures by an isogeometrical approach

A new isogeometrical procedure for optimization of material composition of functionally graded structures in thermo-mechanical processes is introduced. The proposed method employs a generalized form of the standard isogeometric analysis method, allowing for gradation of material properties through patches. The variations of material properties are captured in a fully isogeometric formulation using the same NURBS basis functions employed for construction of the geometry and approximation of the solution. Subsequently, the applicates of control points that define the surfaces of volume fractions of the constituents are considered as the design variables and obtained by solving the optimization problem using a mathematical programming algorithm. Some numerical examples under thermal and mechanical loadings are considered to demonstrate the performance and applicability of the proposed method. Comparison of the obtained results with those of the other existing approaches such as finite elements and meshfree methods verifies the presented results. It will be seen that the proposed procedure considerably removes the difficulties of the existing methods and provides a promising tool for material design of functionally graded structures.

[1]  Ken Brodlie,et al.  Visualization of surface data to preserve positivity and other simple constraints , 1995, Comput. Graph..

[2]  X. Wang,et al.  Non-Linear Response of Functionally Graded Cylindrical Shells under Mechanical and Thermal Loads , 2011 .

[3]  Glaucio H. Paulino,et al.  Optimal design of periodic functionally graded composites with prescribed properties , 2009 .

[4]  F. Bobaru.  Designing Optimal Volume Fractions For Functionally Graded Materials With Temperature-Dependent Material Properties , 2007 .

[5]  Imao Tamura,et al.  Tensile deformation of two-ductile-phase alloys: Flow curves of α-γ FeCrNi alloys , 1976 .

[6]  T. Hughes,et al.  Isogeometric analysis : CAD, finite elements, NURBS, exact geometry and mesh refinement , 2005 .

[7]  Passakorn Vessakosol,et al.  Numerical solutions for functionally graded solids under thermal and mechanical loads using a high-order control volume finite element method , 2011 .

[8]  Z.-H. Jin,et al.  Thermal Fracture of Functionally Graded Plate with Parallel Surface Cracks , 2009 .

[9]  Jin-Rae Cho,et al.  Averaging and finite-element discretization approaches in the numerical analysis of functionally graded materials , 2001 .

[10]  Kyoung Sub Park,et al.  Vibrations of Timoshenko beams with isogeometric approach , 2013 .

[11]  Kyung-Su Na,et al.  Volume fraction optimization of functionally graded composite panels for stress reduction and critical temperature , 2009 .

[12]  Djaffar Boussaa,et al.  Optimization of temperature-dependent functionally graded material bodies , 2009 .

[13]  Zhongming Zheng,et al.  Two-dimensional thermoelasticity solution for functionally graded thick beams , 2006 .

[14]  Behrooz Hassani,et al.  An improved isogeometrical analysis approach to functionally graded plane elasticity problems , 2013 .

[15]  K. Tanaka,et al.  Average stress in matrix and average elastic energy of materials with misfitting inclusions , 1973 .

[16]  D. Gossard,et al.  Synthesis of Optimal Shape and Topology of Structures , 1996 .

[17]  G. Paulino,et al.  ISOPARAMETRIC GRADED FINITE ELEMENTS FOR NONHOMOGENEOUS ISOTROPIC AND ORTHOTROPIC MATERIALS , 2002 .

[18]  Mahmoud Nemat-Alla,et al.  Reduction of thermal stresses by composition optimization of two-dimensional functionally graded materials , 2009 .

[19]  Qi Xia,et al.  Simultaneous optimization of the material properties and the topology of functionally graded structures , 2008, Comput. Aided Des..

[20]  V. E. Beal,et al.  Functionally Graded Materials , 2006 .

[21]  J. Dolbow,et al.  Imposing Dirichlet boundary conditions with Nitsche's method and spline‐based finite elements , 2010 .

[22]  X. Y. Kou,et al.  Optimal design of functionally graded materials using a procedural model and particle swarm optimization , 2012, Comput. Aided Des..

[23]  Dongying Liu,et al.  Free vibration of FGM plates with in-plane material inhomogeneity , 2010 .

[24]  Jin-Rae Cho,et al.  Optimal tailoring of 2D volume-fraction distributions for heat-resisting functionally graded materials using FDM , 2002 .

[25]  S. Vel,et al.  Optimization of natural frequencies of bidirectional functionally graded beams , 2006 .

[26]  S. C. Yen,et al.  Meshless local Petrov-Galerkin analysis for 2D functionally graded elastic solids under mechanical and thermal loads , 2005 .

[27]  T. Rabczuk,et al.  NURBS-based finite element analysis of functionally graded plates: Static bending, vibration, buckling and flutter , 2012, 1210.4676.

[28]  S. Vel,et al.  Two-dimensional optimization of material composition of functionally graded materials using meshless analyses and a genetic algorithm , 2006 .

[29]  Richard Schapery.  Thermal Expansion Coefficients of Composite Materials Based on Energy Principles , 1968 .

[30]  Kyung K. Choi,et al.  Structural Sensitivity Analysis and Optimization 1: Linear Systems , 2005 .

[31]  Hugh A. Bruck,et al.  Enhancing the optimization of material distributions in composite structures using gradient architectures , 2003 .

[32]  Ji-Hwan Kim,et al.  Volume fraction optimization for step-formed functionally graded plates considering stress and critical temperature , 2010 .

[33]  R. Hill.  A self-consistent mechanics of composite materials , 1965 .

[34]  Ryoichi Chiba,et al.  Optimisation of material composition of functionally graded materials based on multiscale thermoelastic analysis , 2012 .

[35]  Jin-Rae Cho,et al.  Volume fraction optimization for minimizing thermal stress in Ni–Al2O3 functionally graded materials , 2002 .

[36]  Z. Zhong,et al.  Two-dimensional thermoelasticity solution for functionally graded thick beams , 2006 .

[37]  Ji-Hwan Kim,et al.  Optimization of volume fractions for functionally graded panels considering stress and critical temperature , 2009 .

[38]  Yoshinari Miyamoto,et al.  Functionally Graded Materials. , 1995 .

[39]  Martin H. Sadd,et al.  Elasticity: Theory, Applications, and Numerics , 2004 .

[40]  Qing Hua Qin,et al.  Meshless approach for thermo-mechanical analysis of functionally graded materials , 2008 .

[41]  Licheng Guo,et al.  An analytical method for thermal stresses of a functionally graded material cylindrical shell under a thermal shock , 2010 .

[42]  R. Xu,et al.  Semi-analytical elasticity solutions for bi-directional functionally graded beams , 2008 .

[43]  Kyung K. Choi,et al.  Structural sensitivity analysis and optimization , 2005 .

[44]  Sergio Turteltaub,et al.  Optimal control and optimization of functionally graded materials for thermomechanical processes , 2002 .

[45]  K. Matsui,et al.  Continuous approximation of material distribution for topology optimization , 2004 .

[46]  S. Vel,et al.  Multi-objective optimization of functionally graded materials with temperature-dependent material properties , 2007 .

[47]  Lin-zhi Wu,et al.  Thermal fracture model for a functionally graded plate with a crack normal to the surfaces and arbitrary thermomechanical properties , 2008 .

[48]  Glaucio H. Paulino,et al.  Design of Functionally Graded Structures Using Topology Optimization , 2005 .

[49]  Carlton Gamer.  Colorado Springs, Colorado , 1973 .

[50]  Les A. Piegl,et al.  The NURBS Book , 1995, Monographs in Visual Communication.

[51]  H. Ding,et al.  Elasticity solutions for plane anisotropic functionally graded beams , 2007 .

[52]  Glaucio H. Paulino,et al.  Layout and material gradation in topology optimization of functionally graded structures: a global–local approach , 2010 .