Layout Optimization For Multiple Materials And Bi-modulus Material In Structures

The purpose of structural optimization is to reduce the cost and improve the properties of the structure which satisfies the specified conditions. Nowadays, topology optimization is the most advanced theory on structural optimization. It has been widely adopted in various of design engineerings. Commonly, the material(s) in a hydraulic structure has two characters, e.g., many materials with different stiffnesses in a structure(e.g., rockfill dam, etc), or the material showing bi-modulus property(e.g., concrete, etc). Bi-modulus material means the moduli of the material under tension and compressioin are different along the same direction. Hence, the elastic tensor of a bi-modulus material is stress dependent, which leads to the reanalysis of a bi-modulus structure for finding the accurate deformation. On the other hand, for a complicated structure such as concrete face rock fill dam(CFRD), it has many types of materials with different stiffnesses. It is hardly to find the optimal materials layout in the strucute using the traditional topology optimization. Meanwhile, the optimal topology of a bi-modulus structure is also difficult to find by the traditional method, effectively. However, it takes a potential risk to use the toplogy optimization result of a bi-modulus structure without considering the bi-modulus property.Considering the difficulties mentioned above, four typical topology optimizations are presented: 1) layout optimization of complicated structure with large number of materials; 2) topology optimization of a structure with a tension-only or compression-only material; 3) topology optimization of a structure with a common bi-modulus material; 4) ill-loaded topology optimization of a continuum. Four optimization methods are proposed to solve the problems. The detailed fruit in research are as follows.(1) A criterion approach, strain energy density(SED) sequencing method, is proposed for multiple materials layout optimization in a complicated continuum. The major idea of the algorithm is to layout the stiffer material in the area with higher SED. Firstly, numbering the materials in structure from higher modulus to lower modulus; secondly, after structural analysis, the areas with specified amount of materials are set to be non-design domain. Ascending order the SEDs of the finite elements in the remained area, i.e., design domain; thirdly, changing the material in some of the lower-SED elements with the neighbored softer material. Finally, iteration stops when all the amount of materials reaches their critical values. The validilty of the present method is proved using many numerical examples and the comparison of the results by both the present method and the solid isotropic microstructure with penalization(SIMP) method. Simultaneously, the effects of such differences on moduli, volume ratios and types of materials on the final layouts are given. Based on the present method, the multiple layout optimization model for a CFRD is built and analyzed. The rules of the materials layout in the dam are investigated for the guidance of practical engineering.(2) The material replacement combining reference interval approach is presented to solve the topology optimization of a continuum with a tension-only or compression-only material, which is a special bi-modulus material. Firstly, the original tension-only or compression-only material is replaced with an isotropic material before the structural deformation analysis in optimization. Secondly, the effective SED of each element are calculated according to the local stress state and the material property. Thirdly, the design variable, i.e., the pseudo-density of each element, is updated according the comparison between the effective SED and the lower and the upper boundaries of reference interval. Finally, the boundaries of the reference interval are updated for finding a feasible solution and the convergency control of algorithm. The validity and the efficiency of algorithm are verified using typical numerical examples from such hydraulic strucutres as beam and bridge. The influence of the tension-only or compression-only property of material on the final topology is discussed. We also discuss the effect of Poisson’s ratio of material on the final layout.(3) The material replacement approach is proposed to find the optimal topology of a common bi-modulus materialin optimization. Using material replacement, the original bi-modulus material is firstly replaced with two isotropic materials with either the tensile modulus or the compressive modulus. Secondly, the tension SED, compression SED and the modification factor of local stiffness are calculated according to the local stress state. The modulus of the isotropic material used in the next structural reanalysis is determined by the maximum of tension SED and compression SED. Thirdly, to have the same local stiffness before and after material replacement, the stiffness matrix of the finite element is modified using the modification factor so as to have the same load transmission path(LTP) in structure. Finally, the pseudo-densities are updated using gradient-based method. The validity and the efficiency of the present method are verified using deep beam and bearin platform numerical examples. Numerical results show that the computational efficiency is almost the same as that for solving isotropic material layout. Secondly, the optimization model for a bi-modulus structural topology optimization under multiple load conditions is built using linear weighted method and the dependence of topology on the difference between the two moduli is analyzed.(4) The fractional-norm objective function method is presented to give efficiently a reasonable design for ill-loaded topology optimizationsuch problem. Firstly, the relation between the major and minor LTP in ill-loaded problem is discussed. Secondly, the definition of the fractional norm of a vector is defined and the combined objective function is the q-normed structural compliances of all loading conditions is defined. The value of norm, i.e., q in(0, 1), is essential for adjusting the contribution of the compliances. The lower value of q will strengthen the minor LTP heavier. Combining the material replacement method, the optimal bi-modulus material layout can be found when the structure is under ill-load conditions. The Numerical results demonstrate that the reasonable topology for supporting the ill-load can be found when q is in [0.1, 0.5], even if the major load is 1000 times of the minor load.From above, the bi-modulus material in the structure behaves different stiffness in different component when solved by the material replacement method. Therefore, the topology optimization of a bi-modulus material can be considered as a pecular multiple materials layout optimization. It lays a foundation for the research of multiple bi-modulus materials layout optimization under complicated loads.