Study On Topology Optimization For Continuum Structure With Multiple Bi-modulus Materials

With nearly three decades of development,nowadays,continuum topology optimization has been used widely in practical design fields,such as aerospace/airspace engineering,mechcanical engineering,composites design engineering,etc.Comparing with sizing/shape optimization,topology optimization behaves difficult in finding the sensitivity of objective function and poor convergence due to large number of design variables.In particular,many types of material with different physical properties may exist simultaneously in a composites/structure.Topology optimization is significant in the design of such material/structure.Hence,topology optimization of multiple materials always attracts much attention and also is a challenge in design fields.When the mateirals are bi-modulus materials,the optimal topology of the materials becomes more difficult.A bi-modulus material implies that it has different tensile and compressive moduli along the same direction.In practical engineering,such materials as concrete,cast iron,glue,rocks and porous media show bi-modulus characteristics.Therefore,the topology optimization of a continuum with many bi-modulus materials is more important in design engineering.To deal this difficulty,several algorithms are proposed in the present study.In finding the optimal topologyies of multiple bi-modulus material in a continuum under complicated loading cases,some research results are obtained.In finding the optimal topology of continuum design domain with multiple materials using a traditional solid isotorpic material with penalization(SIMP)optimization approach,the convergency of the algorithm is poor when a comoplex interplolation scheme is adopted to calculate the equivalent modulus of an element with many mateials.To overcome the difficulty,a new method,called sequencial SIMP(S-SIMP)method,is presented.In the method,the original multiple materials topology optimization problem is replaced with a seires of suboptimization problems.In each suboptimization,there only few of the materials are designable.Four essentials are included in the S-SIMP method.First,all of the(m types)materials in design field are sorted(in ascending order)according to their moduli and the materials are labeled from 1 to m.Second,selecting the first 2~4 materials as designable materials in each suboptimization and solving the subproblem with traditional SIMP method.Third,after the completion of the suboptimization,the material with the largest moduli is set to be nondesignable material,i.e.,the related volume ratio of the stiffest material in each finite element is a constant rather than a design variable,in the next suboptimization.At the same time,a new material is sequentially chosen from the rest materials as deignable material.Final,when all the material has been updated,the optimal solution is obtained.Comparing with the traditional SIMP method,the present method has lower number of design variables,which results in fast convergence of algorithm.Numerical tests demonstrate that the computational cost does not increase obviously.Besides,the final sturcture is a layered structure with clear interfaces among the materials,which is easily manufactured.To find an optimal design of a non layered structure with many materials,we present the inner sequential single solid optimization(ISSSO)method.In the ISSSO method,all the(M types of)materials are sorted according to their moduli and labeled from 1 to M.In the inner iteration,a series of SIMP optimization for single solid(from 1 to M)is carried out.The internal iteration stops when all the materials are updated.Next internal iteration starts if the algorithm does not show convergent.During the inner iteration,the updated materials are considered as nondesignable materials.Thus,the upper limit of the design variable(volume ratio)of a finite element decreases with the increasing of the number of updated materials.Because there is only one solid being updated in each sub optimization,the design variables are updated efficiently and the algorithm convergences easily.The efficiency and effectiveness of the proposed algorithm are verified by numerical examples.Considering the difference between the tensile and the compressive propetrties of a material used in practical engineering,a new topology optimization for multiple bi-modulus materials are presented based on SIMP method.To merge reanalysis of a continuum structure with many bi-modulus materials into the iteration for the design variables in optimization,material replacement scheme is adopted,i.e.,each bi-modulus material is replaced with an isotropic material according to its current stress state.The difference of the local stiffness due to the replacement operation is reduced by modifying the local element stiffness matrix of the finite element.The formula of sensitivities of the objective function and the constraint functions are derived when the objective is to maximize the stiffness of structure.The gradient-based algorithm can be used to update the design variables by the formula.Numerical tests demonstrate that the optimal solution of a continuum with many bi-modulus materials is different from that of the same structure with isotropic materials.The optimal solution depends serious on the ratios between tensile and compressive moduli of materials and the moduli difference among the materials.Especially,the optimal solution is sensitive to the(forward/backward)directions of the loads.The optimal microstructure of a lightweighted material with maximal bulk modulus can be found using topology optimization.A layered lightweighted sandwich structure can be fabricated using the microstructure.The stiffness/strength-to-density ratios of the layered structure depend on both of the configuration of the core and the manufacturing process.In the present study,a sandwich structure with adhesive corrugated cores is investigated in finding the damage mechanism of the structure with weaker bonding property.In practical engineering,the thickness of the adhesive layer is far less than the sizes of a finite element in the discretized structure.Hence,the original continuum adhesive layers are discretized with beam elements.Due to the weak strength of the adhesive layer,the strength criterion of the adhesive layer(beam element)is proposed.For a broken beam element in the adhesive layer,its stiffness depends on the stress state,i.e.,the tensile and compressive properties of the broken beam elements are different.Hence,the reduction factor for the stiffness of the broken beam element is calculated.A sandwich beam structure under three point bending deformation is studied in numerical tests.The relations between the damage of the structure and the configurations of the cores(shape of corrugated core and the overlap ratios between cores and panels)are tested and the optimal microstructure of the cores is given when the beam structure has the maximal strength-to-density ratio.Practically,most of the structures are under multiple loading cases.In the present study,the topology optimization of multi phase bi-modulus materials under multiple loading cases are under investigation.Firstly,we study the characteristics of the optimal solutions of a continuum with many isotropic materials under multiple loading cases using ISSSO method.Secondly,a general objective function is formed by linear weighted scheme on the loading cases and the topology optimization of the continuum with many bi-modulus materials is solved numerically.Numerical results indicate that the final layout of the materials is sensitive to such factors as loading cases,moduli difference among materials and the ratios of the tensile and compressive moduli of the bi-modulus materials.Finally,the optimal solution of the multiple bimodulus material under ill-loaded cases is researched.The fraction-normed objective function is adopted.And the material replacement scheme is adopted for dealing with bi-modulus problem.To reduce the error of stiffness of structure,the modification factor of the local stiffness is calculated according to the local strain energy density.And the sensitivity formula of the objective function is derived with consideration of the modification factor.The solutions with respect to different values of the norm(Q)are studied.Numerical results demonstrate that the reasonable solution of the structure under heavy ill-load cases(the strongest load is over 100 times of the weakest)can be found when Q is chosen from the interval [0.2,0.5].