topology optimization of hyperelastic structures with frictionless contact supports abstract hyperelastic structures usually undergo large deformations and thus may be subject to deformation dependent contact supports this paper presents an effective topology optimization methodology for the compliance minimization design of hyperelastic structures with frictionless contact supports in the optimization model the strain energy function of hyperelastic material is represented by an artificial penalization model and the contact boundary conditions are modeled with hypothetical nonlinear springs the additive hyperelasticity technique is employed for circumventing the local buckling instability exhibited by low density elements in conjunction with the adjoint variable sensitivity analysis the nonlinear topology optimization problem is solved by a gradient based mathematical programming algorithm numerical examples are given to show the importance of considering contact supports and to demonstrate the applicability of the proposed method

achieving minimum length scale in topology optimization using nodal design variables and projection functions a methodology for imposing a minimum length scale on structural members in discretized topology optimization problems is described nodal variables are implemented as the design variables and are projected onto element space to determine the element volume fractions that traditionally define topology the projection is made via mesh independent functions that are based upon the minimum length scale a simple linear projection scheme and a non linear scheme using a regularized heaviside step function to achieve nearly 0 1 solutions are examined the new approach is demonstrated on the minimum compliance problem and the popular simp method is used to penalize the stiffness of intermediate volume fraction elements solutions are shown to meet user defined length scale criterion without additional constraints penalty functions or sensitivity filters no instances of mesh dependence or checkerboard patterns have been observed copyright 2004 john wiley sons ltd

numerical methods for the topology optimization of structures that exhibit snap through the inclusion of non linear elastic analyses into the topology optimization problem is necessary to capture the finite deformation response e g the geometric non linear response of compliant mechanisms in previous work the non linear response is computed by standard non linear elastic finite element analysis here we incorporate a load displacement constraint method to traverse non linear equilibrium paths with limit points to design structures that exhibit snap through behaviour to accomplish this we modify the basic arc length algorithm and embed this analysis into the topology optimization problem copyright 2002 john wiley sons ltd

topology optimization of an acoustic horn we present a method for topology optimization of an acoustic horn with the aim of radiating sound as efficiently as possible using a strategy commonly employed for topology optimization of elastic structures we optimize over a scalar function indicating presence of material the helmholtz equation modeling the wave propagation is solved using the finite element method and the associated adjoint equation provides the required gradients numerical experiments validates that the result of the optimization provides horns with the desired acoustical properties the resulting horns are very efficient in the frequency span subject to optimization

a newly developed qp relaxation method for element connectivity parameterization to achieve stress based topology optimization for geometrically nonlinear structures the aim of this work is to present a novel computational approach to employ the stress based topology optimization method stom to minimize the volume subject to the locally defined stress constraints of a geometrically nonlinear structure in the framework of the element connectivity parameterization ecp method considering the locally defined stress constraints in topology optimization to is a classic and challenging engineering problem and successful optimization procedures have recently been developed using the density based to method for linear elastic structures however no study has yet considered the static failure constraint when using to for a geometrically nonlinear structure therefore the present study develops a novel computational approach for the stom for a geometrically nonlinear structure to successfully optimize a geometrically nonlinear structure the unstable element issue must be properly addressed in addition to the stress singularity issue the existence of a large number of constraints and the highly nonlinear behavior of the local stress constraints to effectively resolve these issues this research adopts the ecp method to interpolate and optimize the connectivities among solid finite elements furthermore we find that a stress singularity issue linked to the local optima issue arises in the ecp method that is different from that of the density based to by investigating the singularity behavior in detail we develop a new qp relaxation method that is suitable for the ecp method to demonstrate the improved capability of the proposed ecp method with the modified qp relaxation several two dimensional to problems are solved

element connectivity parameterization for topology optimization of geometrically nonlinear structures in the current element density based topology optimization element stiffness is penalized to yield either solid or void or very weak materials but low density elements appear during and even after optimization iterations especially when nonlinear problems involving large deformation are considered low density finite elements cause serious numerical problems as their tangent stiffness matrices lose positive definiteness to completely eliminate the problem caused by low density elements we propose a new method all finite elements are kept solid throughout the optimization process zero length elastic links are introduced to parameterize inter element connectivity the link stiffness is penalized in this approach the design variables are defined on the links and vary from 0 and 1 corresponding to the unconnected and rigidly connected states since the finite elements used to discretize the analysis domain always remain solid typical numerical problems encountered by the standard element density based formulation disappear to implement the present method several issues such as the handling of the mass constraint and raster imaging are also investigated

development of pareto topology optimization considering thermal loads abstract in this research we developed a level set based topology optimization with a topological derivative formulation considering thermal load thermo elasticity equations were utilized to obtain the sensitivity of the objective function after inserting a small hole in the domain total strain energy and the maximum stress in the design domain were taken as the objective functions after taking the thermal loading effect into account the total strain energy density function became a nonhomogeneous function of the strain in addition temperature variation changed hooke s law from a linear homogeneous to a linear nonhomogeneous expression including a zero order term we derived the sensitivity value of the selected objective functions with respect to a perturbation in the structural domain under mechanical and thermal loads while considering these changes in the governing equations we performed several numerical optimization problems to demonstrate the validity of the present level set based pareto topology optimization two types of examples compliance and stress minimization were solved based on the chosen objective functions furthermore in the stress minimization examples the derived formula was extended to consider thermal effects in the failure theories for pressure independent and dependent materials

a monolithic approach for topology optimization of electrostatically actuated devices abstract this paper presents a new topology optimization scheme for nonlinear electrostatic systems actuated by coulomb s forces for successful optimization computational issues such as i alternating governing equations with respect to spatially defined design variables ii imposing interaction boundary conditions between insulator air and conductor solid and iii control of minimum geometrical feature sizes must be addressed to address the first two issues the paper presents a monolithic formulation based on continuum mechanics theory which simultaneously calculates the electric potential and structural displacements to interpolate between insulator and conductor with continuous design variables the monolithic approach distinguishes between the permittivity value of the electric poisson s equation and that of maxwell s stress tensor for the optimization standard material interpolations are used for young s modulus and permittivity values moreover a recently developed morphology filter is applied to control electrode gaps and other geometrical features

on some universal algebras associated to the category of lie bialgebras in our previous work math 0008128 we studied the set quant k of all universal quantization functors of lie bialgebras over a field k of characteristic zero compatible with the operations of taking duals and doubles we showed that quant k is canonically isomorphic to a product g0 k k where g0 k is a universal group and k is a quotient set of a set b k of families of lie polynomials by the action of a group g k we prove here that g0 k is equal to the multiplicative group 1 k so quant k is as close as it can be to k we also prove that the only universal derivations of lie bialgebras are multiples of the composition of the bracket with the cobracket finally we prove that the stabilizer of any element of b k is reduced to the 1 parameter subgroup generated by the corresponding square of the antipode

optimal layout design of three dimensional geometrically non linear structures using the element connectivity parameterization method the topology design optimization of three dimensional geometrically non linear continuum structures is still difficult not only because of the size of the problem but also because of the unstable continuum finite elements that arise during the optimization to overcome these difficulties the element connectivity parameterization ecp method with two implementation formulations is proposed in ecp structural layouts are represented by inter element connectivity which is controlled by the stiffness of element connecting zero length links depending on the link location ecp may be classified as an external ecp e ecp or an internal ecp i ecp in this paper i ecp is newly developed to substantially enhance computational efficiency the main idea in i ecp is to reduce system matrix size by eliminating some internal degrees of freedom associated with the links at voxel level as for ecp implementation with commercial software e ecp developed earlier for two dimensional problems is easier to use even for three dimensional problems because it requires only numerical analysis results for design sensitivity calculation the characteristics of the i ecp and e ecp methods are compared and these methods are validated with numerical examples copyright 2006 john wiley sons ltd

a mixed fem approach to stress constrained topology optimization we present an alternative topology optimization formulation capable of handling the presence of stress constraints in a straightforward fashion the main idea is to adopt a mixed finite element discretization scheme wherein not only displacements as usual but also stresses are the variables entering the formulation by doing so any stress constraint may be handled within the optimization procedure without resorting to post processing operation typical of displacement based techniques that may also cause a loss in accuracy in stress computation if no smoothing of the stress is performed two dual variational principles of hellinger reissner type are presented in continuous and discrete form that which included in a rather general topology optimization problem in the presence of stress constraints that is solved by the method of moving asymptotes int j numer meth engng 1984 24 3 359 373 extensive numerical simulations are performed and ongoing extensions outlined including the optimization of elastoplastic and incompressible media copyright 2007 john wiley sons ltd

uniqueness of coxeter structures on kac moody algebras abstract let g be a symmetrizable kac moody algebra and u g the corresponding quantum group we showed in 1 2 that the braided coxeter structure on integrable category o representations of u g which underlies the r matrix actions arising from the levi subalgebras of u g and the quantum weyl group action of the generalized braid group b g can be transferred to integrable category o representations of g we prove in this paper that up to unique equivalence there is a unique such structure on the latter category with prescribed restriction functors r matrices and local monodromies this extends simplifies and strengthens a similar result of the second author valid when g is semisimple and is used in 3 to describe the monodromy of the rational casimir connection of g in terms of the quantum weyl group operators of u g our main tool is a refinement of enriquez s universal algebras which is adapted to the prop describing a lie bialgebra graded by the non negative roots of g

topology optimization of acoustic structure interaction problems using a mixed finite element formulation the paper presents a gradient based topology optimization formulation that allows to solve acoustic structure vibro acoustic interaction problems without explicit boundary interface representation in acoustic structure interaction problems the pressure and displacement fields are governed by helmholtz equation and the elasticity equation respectively normally the two separate fields are coupled by surface coupling integrals however such a formulation does not allow for free material re distribution in connection with topology optimization schemes since the boundaries are not explicitly given during the optimization process in this paper we circumvent the explicit boundary representation by using a mixed finite element formulation with displacements and pressure as primary variables a u p formulation the helmholtz equation is obtained as a special case of the mixed formulation for the elastic shear modulus equating to zero hence by spatial variation of the mass density shear and bulk moduli we are able to solve the coupled problem by the mixed formulation using this modelling approach the topology optimization procedure is simply implemented as a standard density approach several two dimensional acoustic structure problems are optimized in order to verify the proposed method copyright 2006 john wiley sons ltd

stress based shape and topology optimization with the level set method abstract this paper proposes a level set method to solve minimum stress and stress constrained shape and topology optimization problems the method solves a sub optimization problem every iteration to obtain optimal boundary velocities a p norm stress functional is used to aggregate stresses in a single constraint the shape sensitivity function is derived and a computational procedure based on a least squares interpolation approach is devised in order to compute sensitivities at the boundaries adaptive constraint scaling is used to enforce exact control of stress limits numerical results show that the method is able to solve the problem efficiently for single and multiple load cases obtaining solutions with smooth boundaries

topology optimization of continuum structures with local stress constraints we introduce an extension of current technologies for topology optimization of continuum structures which allows for treating local stress criteria we first consider relevant stress criteria for porous composite materials initially by studying the stress states of the so called rank 2 layered materials then on the basis of the theoretical study of the rank 2 microstructures we propose an empirical model that extends the power penalized stiffness model also called simp for solid isotropic microstructure with penalization for inter mediate densities in a second part solution aspects of topology problems are considered to deal with the so called singularity phenomenon of stress constraints in topology design an constraint relaxation of the stress constraints is used we describe the mathematical programming approach that is used to solve the numerical optimization problems and show results for a number of example applications 1998 john wiley sons ltd

stress based topology optimization method for steady state fluid structure interaction problems abstract this research developed a new stress based topology optimization method stom for a steady state fluid structure interaction fsi problem that minimizes the volume subject to the local stress constraints despite numerous studies on stom challenging optimization issues related to stress based topology optimization to procedures for fluid structure multiphysics systems still exist critical issues involved in creating a successful to for an fsi structure include the interpolation approach between the fluid equation and the structure equation with respect to locally defined design variables the mutual multiphysics coupling boundary conditions at dramatically evolving interfacing boundaries and a clear interpretation of the governing equations and the interaction boundary conditions for spatially varying intermediate design variables in addition to these three issues which are related to multiphysics equations there are three important considerations related to the stom the stress singularity issue the issues of multiple constraints and the highly nonlinear behavior of the stress constraints to resolve all of the aforementioned issues we applied a monolithic analysis integrating the qp relaxation method and the global p norm approach using the present method we created optimal layouts that minimize the volume constraining local stress values for a steady state fluid and structural interaction system

topology optimization of creeping fluid flows using a darcy stokes finite element a new methodology is proposed for the topology optimization of fluid in stokes flow the binary design variable and no slip condition along the solid fluid interface are regularized to allow for the use of continuous mathematical programming techniques the regularization is achieved by treating the solid phase of the topology as a porous medium with flow governed by darcy s law fluid flow throughout the design domain is then expressed as a single system of equations created by combining and scaling the stokes and darcy equations the mixed formulation of the new darcy stokes system is solved numerically using existing stabilized finite element methods for the individual flow problems convergence to the no slip condition is demonstrated by assigning a low permeability to solid phase and results suggest that auxiliary boundary conditions along the solid fluid interface are not needed the optimization objective considered is to minimize dissipated power and the technique is used to solve examples previously examined in literature the advantages of the darcy stokes approach include that it uses existing stabilization techniques to solve the finite element problem it produces 0 1 void solid topologies i e there are no regions of artificial material and that it can potentially be used to optimize the layout of a microscopically porous material copyright 2005 john wiley sons ltd

phase field relaxation of topology optimization with local stress constraints we introduce a new relaxation scheme for structural topology optimization problems with local stress constraints based on a phase field method in the basic formulation we have a pde constrained optimization problem where the finite element and design analysis are solved simultaneously the starting point of the relaxation is a reformulation of the material problem involving linear and 0 1 constraints only the 0 1 constraints are then relaxed and approximated by a cahn hilliard type penalty in the objective functional which yields convergence of minimizers to 0 1 designs as the penalty parameter decreases to zero a major advantage of this kind of relaxation opposed to standard approaches is a uniform constraint qualification that is satisfied for any positive value of the penalization parameter the relaxation scheme yields a large scale optimization problem with a high number of linear inequality constraints we discretize the problem by finite elements and solve the arising finite dimensional programming problems by a primal dual interior point method numerical experiments for problems with local stress constraints based on different criteria indicate the success and robustness of the new approach

brittle and ductile failure constraints of stress based topology optimization method for fluid structure interactions abstract this study considers failure theories for brittle and ductile materials in the stress based topology optimization method stom for steady state fluid structure interactions fsi in some relevant studies the subject of the stress based topology optimization to minimize volumes with local von mises stress constraints has been researched however the various failure theories for ductile and brittle materials such as the maximum shear stress theory the brittle and ductile mohr coulomb theory and the drucker prager theory have not been considered for successful stom for fsi in addition to alleviating physics interpolation issues between structure and fluid and some numerical issues related to stom the mathematical characteristics of the various failure theories should be properly formulated and constrained to resolve all the involved computational issues the present study applies the monolithic analysis method the qp relaxation method and the p norm approach to the failure constraints the present topology optimization method can create optimal layouts while minimizing volume constraining local failure constraints for ductile and brittle materials for steady state fluid and structural interaction system

topology optimization with local stress constraint based on level set evolution via reaction diffusion abstract this work focuses the structural topology optimization problem of mass minimization subject to local stress constraints to this aim two related issues are addressed the first one is the successful strategy used to define local stress constraints by means of an augmented lagrangian approach the second and main contribution of the present paper is the use of a reaction diffusion equation to guide via evolution of a level set the design optimization sequence the advantages of this strategy are twofold firstly it allows the creation of new holes during the optimization process a significant feature for a true topological optimization method secondly reinitialization steps usually found in classical hamilton jacobi based evolution are eliminated with a significant improvement in convergence ease a set of benchmark examples in two dimensions are presented numerical results show the efficiency of the algorithm to create new holes identify stress concentrations and to provide stable optimization sequences converging to local minima defined by stress saturated designs

topology optimization for stationary fluid structure interaction problems using a new monolithic formulation this paper outlines a new procedure for topology optimization in the steady state fluid structure interaction fsi problem a review of current topology optimization methods highlights the difficulties in alternating between the two distinct sets of governing equations for fluid and structure dynamics hereafter the fluid and structural equations respectively and in imposing coupling boundary conditions between the separated fluid and solid domains to overcome these difficulties we propose an alternative monolithic procedure employing a unified domain rather than separated domains which is not computationally efficient in the proposed analysis procedure the spatial differential operator of the fluid and structural equations for a deformed configuration is transformed into that for an undeformed configuration with the help of the deformation gradient tensor for the coupling boundary conditions the divergence of the pressure and the darcy damping force are inserted into the solid and fluid equations respectively the proposed method is validated in several benchmark analysis problems topology optimization in the fsi problem is then made possible by interpolating young s modulus the fluid pressure of the modified solid equation and the inverse permeability from the damping force with respect to the design variables

design sensitivity analysis and topology optimization of displacement loaded non linear structures abstract a continuum based design sensitivity analysis dsa method for geometrically nonlinear systems with nonhomogeneous boundary conditions is developed to topologically optimize the displacement loaded nonlinear structures in the adjoint variable method the solution space requires just homogeneous boundary conditions even if the original system has nonhomogeneous ones a design sensitivity expression for the instantaneous rigidity functional is derived for the displacement loaded nonlinear topology optimization the tangent stiffness is obtained at the end of the equilibrium iterations in the nonlinear analysis of the original system this stiffness is used in the dsa so that no iteration would be necessary to evaluate the design sensitivity expressions in force loaded systems the solution dose not converge easily because the material distributes sparsely sometimes during optimization however when the displacement loaded system is used there is no convergence difficulty

topology optimization of geometrically nonlinear structures based on an additive hyperelasticity technique abstract this paper presents a simple but effective additive hyperelasticity technique to circumvent numerical difficulties in solving the material density based topology optimization of elastic structures undergoing large displacements by adding a special hyperelastic material to the design domain excessive distortion and numerical instability occurred in the low density or intermediate density elements are thus effectively alleviated during the optimization process the properties of the additional hyperelastic material are established based on a new interpolation scheme which allows the nonlinear mechanical behaviour of the remodelled structure to achieve an acceptable approximation to the original structure in conjunction with the adjoint variable scheme for sensitivity analysis the topology optimization problem is solved by a gradient based mathematical programming algorithm numerical examples are given to demonstrate the effectiveness of the proposed method

the method of moving asymptotes a new method for structural optimization a new method for non linear programming in general and structural optimization in particular is presented in each step of the iterative process a strictly convex approximating subproblem is generated and solved the generation of these subproblems is controlled by so called moving asymptotes which may both stabilize and speed up the convergence of the general process

theoretical aspects of the internal element connectivity parameterization approach for topology optimization the internal element connectivity parameterization i ecp method is an alternative approach to overcome numerical instabilities associated with low stiffness element states in non linear problems in i ecp elements are connected by zero length links while their link stiffness values are varied therefore it is important to interpolate link stiffness properly to obtain stably converging results the main objective of this work is two fold 1 the investigation of the relationship between the link stiffness and the stiffness of a domain discretizing patch by using a discrete model and a homogenized model and 2 the suggestion of link stiffness interpolation functions the effects of link stiffness penalization on solution convergence are then tested with several numerical examples the developed homogenized i ecp model can also be used to physically interpret an intermediate design variable state

triangular checkerboard control using a wavelet based method in topology optimization topology optimization has been carried out mainly with structured quadrilateral finite elements however triangular elements facilitate mesh generation especially for problems having complex geometries that often appear in practical industrial problems the use of triangular elements especially low order triangular elements causes a serious numerical trouble that is equivalent to the rectangular checkerboard pattern formation the objective of this investigation is to develop a triangular checkerboard freeing method that directly restricts the design space to this end we use the multiscale design space that is mapped from the standard single scale density space to facilitate the mapping we employ the triangular mesh subdivision and propose a bi orthogonal wavelet transform suitable for a triangulated domain for checkerboard free designs a shrinkage method based on the wavelet frame appropriate for triangular mesh is proposed typical benchmark problems and a simplified roof reinforcing problem in an automobile body are considered to check the effectiveness of the proposed method copyright 2005 john wiley sons ltd