Topology Optimization (topology + optimization)

Distribution by Scientific Domains


Selected Abstracts


Topology optimization for stationary fluid,structure interaction problems using a new monolithic formulation

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 5 2010
Gil Ho Yoon
Abstract 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. Copyright 2009 John Wiley & Sons, Ltd. [source]


Topology optimization of muffler internal partitions for improving acoustical attenuation performance

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 4 2009
Jin Woo Lee
Abstract The internal partition configuration of an expansion chamber muffler affects significantly its acoustical transmission characteristics, but the use of systematic optimization methods to muffler design problems is rare. The main objective of this research is to maximize the transmission loss at target frequencies by optimizing partition layouts inside a muffler chamber by formulating an acoustical topology optimization problem. The selected target frequencies include the deep frequencies of a nominal muffler in order to see the critical effects of partition configurations on the acoustical transmission characteristics. The effects of partition volume constraint ratios are also investigated and physics behind the optimized layouts is investigated. Numerical results show that mufflers with optimized partition layouts outperform nominal mufflers considerably, but the shapes and locations of the optimized partitions should be much different from those of conventional partitions. Copyright 2009 John Wiley & Sons, Ltd. [source]


Topology optimization by a neighbourhood search method based on efficient sensitivity calculations

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 12 2006
K. Svanberg
Abstract This paper deals with topology optimization of discretized load-carrying continuum structures, where the design of the structure is represented by binary design variables indicating material or void in the various finite elements. Efficient exact methods for discrete sensitivity calculations are developed. They utilize the fact that if just one or two binary variables are changed to their opposite binary values then the new stiffness matrix is essentially just a low-rank modification of the old stiffness matrix, even if some nodes in the structure may disappear or re-enter. As an application of these efficient sensitivity calculations, a new neighbourhood search method is presented, implemented, and applied on some test problems, one of them with 6912 nine-node finite elements where the von Mises stress in each non-void element is considered. Copyright 2006 John Wiley & Sons, Ltd. [source]


On optimization of bio-probes

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 6 2004
N. L. Pedersen
Abstract The present paper deals with the modelling and optimization of small bio-probes that can be used for biological sensing; the bio-probes can be classified as MicroElectroMechnical Systems (MEMS). The objective is to optimize the structure of the bio-probes in order to maximize the sensing sensitivity. A biological coating results in a prestress on the sensing cantilever when certain molecules are present in the surrounding medium. The mechanical deformation due to the biological material is modelled by applying a prestress in the top layer of the bio-probes. Topology optimization is used to improve the design. In the present work it is necessary to use an interpolation scheme different from the SIMP (power law) approach which is usually used in topology optimization. In calculating the sensitivities, needed for the optimization, complications due to the prestress occur, but also due to the coupling between the elastic field and the electric field which both must be used in an integrated model. These complications are dealt with and analytically obtained sensitivities are presented. Copyright 2004 John Wiley & Sons, Ltd. [source]


Topology optimization of microfluidic mixers

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 5 2009
Casper Schousboe Andreasen
Abstract This paper demonstrates the application of the topology optimization method as a general and systematic approach for microfluidic mixer design. The mixing process is modeled as convection dominated transport in low Reynolds number incompressible flow. The mixer performance is maximized by altering the layout of flow/non-flow regions subject to a constraint on the pressure drop between inlet and outlet. For a square cross-sectioned pipe the mixing is increased by 70% compared with a straight pipe at the cost of a 2.5 fold increase in pressure drop. Another example where only the bottom profile of the channel is a design domain results in intricate herring bone patterns that confirm findings from the literature. Copyright 2008 John Wiley & Sons, Ltd. [source]


A continuous adjoint formulation for the computation of topological and surface sensitivities of ducted flows

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 8 2008
C. Othmer
Abstract Topology optimization of fluid dynamic systems is a comparatively young optimal design technique. Its central ingredient is the computation of topological sensitivity maps. Whereas, for finite element solvers, implementations of such sensitivity maps have been accomplished in the past, this study focuses on providing this functionality within a professional finite volume computational fluid dynamics solver. On the basis of a continuous adjoint formulation, we derive the adjoint equations and the boundary conditions for typical cost functions of ducted flows and present first results for two- and three-dimensional geometries. Emphasis is placed on the versatility of our approach with respect to changes in the objective function. We further demonstrate that surface sensitivity maps can also be computed with the implemented functionality and establish their connection with topological sensitivities. Copyright 2008 John Wiley & Sons, Ltd. [source]


A note on hinge-free topology design using the special triangulation of design elements

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 12 2005
Jae Eun Kim
Abstract Hinges, the unrepeated checkerboard cells, may appear in the topology optimization using low-order finite elements, especially for compliant mechanism design. Existing hinge-controlling methods are based on the rectangular element discretization, so slant or curved boundary lines may not be represented satisfactorily. To avoid hinge formation and to represent curved boundary lines better, we consider a macro-design element method which subdivides the design element into eight triangular finite elements; the finite element calculation is carried out with triangular elements, but the design variables are defined at the nodes defining rectangular macro-design elements. For hinge-free results, different stiffness interpolations are suggested depending on whether the triangular element belongs to a master group or a slave group. The performance of the proposed method was checked with compliant mechanism design problems from the viewpoint of hinge suppression and the possibility of generating slant boundary lines. Copyright 2005 John Wiley & Sons, Ltd. [source]


A mixed integer programming for robust truss topology optimization with stress constraints

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 13 2010
Yoshihiro Kanno
Abstract This paper presents a mixed integer programming (MIP) formulation for robust topology optimization of trusses subjected to the stress constraints under the uncertain load. A design-dependent uncertainty model of the external load is proposed for dealing with the variation of truss topology in the course of optimization. For a truss with the discrete member cross-sectional areas, it is shown that the robust topology optimization problem can be reduced to an MIP problem, which is solved globally. Numerical examples illustrate that the robust optimal topology of a truss depends on the magnitude of uncertainty. Copyright 2010 John Wiley & Sons, Ltd. [source]


Polygonal finite elements for topology optimization: A unifying paradigm

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 6 2010
Cameron Talischi
Abstract In topology optimization literature, the parameterization of design is commonly carried out on uniform grids consisting of Lagrangian-type finite elements (e.g. linear quads). These formulations, however, suffer from numerical anomalies such as checkerboard patterns and one-node connections, which has prompted extensive research on these topics. A problem less often noted is that the constrained geometry of these discretizations can cause bias in the orientation of members, leading to mesh-dependent sub-optimal designs. Thus, to address the geometric features of the spatial discretization, we examine the use of unstructured meshes in reducing the influence of mesh geometry on topology optimization solutions. More specifically, we consider polygonal meshes constructed from Voronoi tessellations, which in addition to possessing higher degree of geometric isotropy, allow for greater flexibility in discretizing complex domains without suffering from numerical instabilities. Copyright 2009 John Wiley & Sons, Ltd. [source]


Topology optimization for stationary fluid,structure interaction problems using a new monolithic formulation

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 5 2010
Gil Ho Yoon
Abstract 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. Copyright 2009 John Wiley & Sons, Ltd. [source]


Reducing dimensionality in topology optimization using adaptive design variable fields

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 8 2010
James K. Guest
Abstract Topology optimization methodologies typically use the same discretization for the design variable and analysis meshes. Analysis accuracy and expense are thus directly tied to design dimensionality and optimization expense. This paper proposes leveraging properties of the Heaviside projection method (HPM) to separate the design variable field from the analysis mesh in continuum topology optimization. HPM projects independent design variables onto element space over a prescribed length scale. A single design variable therefore influences several elements, creating a redundancy within the design that can be exploited to reduce the number of independent design variables without significantly restricting the design space. The algorithm begins with sparse design variable fields and adapts these fields as the optimization progresses. The technique is demonstrated on minimum compliance (maximum stiffness) problems solved using continuous optimization and genetic algorithms. For the former, the proposed algorithm typically identifies solutions having objective functions within 1% of those found using full design variable fields. Computational savings are minor to moderate for the minimum compliance formulation with a single constraint, and are substantial for formulations having many local constraints. When using genetic algorithms, solutions are consistently obtained on mesh resolutions that were previously considered intractable. Copyright 2009 John Wiley & Sons, Ltd. [source]


Piecewise constant level set method for structural topology optimization

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 4 2009
Peng Wei
Abstract In this paper, a piecewise constant level set (PCLS) method is implemented to solve a structural shape and topology optimization problem. In the classical level set method, the geometrical boundary of the structure under optimization is represented by the zero level set of a continuous level set function, e.g. the signed distance function. Instead, in the PCLS approach the boundary is described by discontinuities of PCLS functions. The PCLS method is related to the phase-field methods, and the topology optimization problem is defined as a minimization problem with piecewise constant constraints, without the need of solving the Hamilton,Jacobi equation. The result is not moving the boundaries during the iterative procedure. Thus, it offers some advantages in treating geometries, eliminating the reinitialization and naturally nucleating holes when needed. In the paper, the PCLS method is implemented with the additive operator splitting numerical scheme, and several numerical and procedural issues of the implementation are discussed. Examples of 2D structural topology optimization problem of minimum compliance design are presented, illustrating the effectiveness of the proposed method. Copyright 2008 John Wiley & Sons, Ltd. [source]


A dual algorithm for the topology optimization of non-linear elastic structures

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 4 2009
C. S. Jog
Abstract Dual algorithms are ideally suited for the purpose of topology optimization since they work in the space of Lagrange multipliers associated with the constraints. To date, dual algorithms have been applied only for linear structures. Here we extend this methodology to the case of non-linear structures. The perimeter constraint is used to make the topology problem well-posed. We show that the proposed algorithm yields a value of perimeter that is close to that specified by the user. We also address the issue of manufacturability of these designs, by proposing a variant of the standard dual algorithm, which generates designs that are two-dimensional although the loading and the geometry are three-dimensional. Copyright 2008 John Wiley & Sons, Ltd. [source]


Theoretical aspects of the internal element connectivity parameterization approach for topology optimization

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 6 2008
Gil Ho Yoon
Abstract 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. Copyright 2008 John Wiley & Sons, Ltd. [source]


A mixed FEM approach to stress-constrained topology optimization

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 12 2008
M. Bruggi
Abstract 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. [source]


Large-scale topology optimization using preconditioned Krylov subspace methods with recycling

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 12 2007
Shun Wang
Abstract The computational bottleneck of topology optimization is the solution of a large number of linear systems arising in the finite element analysis. We propose fast iterative solvers for large three-dimensional topology optimization problems to address this problem. Since the linear systems in the sequence of optimization steps change slowly from one step to the next, we can significantly reduce the number of iterations and the runtime of the linear solver by recycling selected search spaces from previous linear systems. In addition, we introduce a MINRES (minimum residual method) version with recycling (and a short-term recurrence) to make recycling more efficient for symmetric problems. Furthermore, we discuss preconditioning to ensure fast convergence. We show that a proper rescaling of the linear systems reduces the huge condition numbers that typically occur in topology optimization to roughly those arising for a problem with constant density. We demonstrate the effectiveness of our solvers by solving a topology optimization problem with more than a million unknowns on a fast PC. Copyright 2006 John Wiley & Sons, Ltd. [source]


Topology optimization by a neighbourhood search method based on efficient sensitivity calculations

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 12 2006
K. Svanberg
Abstract This paper deals with topology optimization of discretized load-carrying continuum structures, where the design of the structure is represented by binary design variables indicating material or void in the various finite elements. Efficient exact methods for discrete sensitivity calculations are developed. They utilize the fact that if just one or two binary variables are changed to their opposite binary values then the new stiffness matrix is essentially just a low-rank modification of the old stiffness matrix, even if some nodes in the structure may disappear or re-enter. As an application of these efficient sensitivity calculations, a new neighbourhood search method is presented, implemented, and applied on some test problems, one of them with 6912 nine-node finite elements where the von Mises stress in each non-void element is considered. Copyright 2006 John Wiley & Sons, Ltd. [source]


Structural topology optimization of vibrating structures with specified eigenfrequencies and eigenmode shapes

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 5 2006
Y. Maeda
Abstract In vibration optimization problems, eigenfrequencies are usually maximized in the optimization since resonance phenomena in a mechanical structure must be avoided, and maximizing eigenfrequencies can provide a high probability of dynamic stability. However, vibrating mechanical structures can provide additional useful dynamic functions or performance if desired eigenfrequencies and eigenmode shapes in the structures can be implemented. In this research, we propose a new topology optimization method for designing vibrating structures that targets desired eigenfrequencies and eigenmode shapes. Several numerical examples are presented to confirm that the method presented here can provide optimized vibrating structures applicable to the design of mechanical resonators and actuators. Copyright 2006 John Wiley & Sons, Ltd. [source]


On the convergence of stationary sequences in topology optimization

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 1 2005
Anton Evgrafov
Abstract We consider structural topology optimization problems including unilateral constraints arising from non-penetration conditions in contact mechanics. The resulting non-convex non-smooth problems are instances of mathematical programs with equilibrium constraints (MPEC), or bi-level programs. Applying nested (implicit programming) algorithms to this class of problems is problematic owing to the singularity of the feasible set. We propose a perturbation strategy combining the relaxation of the equilibrium constraint with the restriction of the design domain to its regular part only. This strategy allows us to attack the problem numerically using standard non-linear programming algorithms. We rigorously study the optimality conditions for the original singular problem as well as the convergence of stationary points and globally optimal solutions to approximating problems towards respective stationary points and globally optimal solutions to the original problem. A limited numerical benchmarking of the algorithm is performed. Copyright 2005 John Wiley & Sons, Ltd. [source]


On optimization of bio-probes

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 6 2004
N. L. Pedersen
Abstract The present paper deals with the modelling and optimization of small bio-probes that can be used for biological sensing; the bio-probes can be classified as MicroElectroMechnical Systems (MEMS). The objective is to optimize the structure of the bio-probes in order to maximize the sensing sensitivity. A biological coating results in a prestress on the sensing cantilever when certain molecules are present in the surrounding medium. The mechanical deformation due to the biological material is modelled by applying a prestress in the top layer of the bio-probes. Topology optimization is used to improve the design. In the present work it is necessary to use an interpolation scheme different from the SIMP (power law) approach which is usually used in topology optimization. In calculating the sensitivities, needed for the optimization, complications due to the prestress occur, but also due to the coupling between the elastic field and the electric field which both must be used in an integrated model. These complications are dealt with and analytically obtained sensitivities are presented. Copyright 2004 John Wiley & Sons, Ltd. [source]


Reduced modified quadratures for quadratic membrane finite elements

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 6 2004
Craig S. Long
Abstract Reduced integration is frequently used in evaluating the element stiffness matrix of quadratically interpolated finite elements. Typical examples are the serendipity (Q8) and Lagrangian (Q9) membrane finite elements, for which a reduced 2 2 Gauss,Legendre integration rule is frequently used, as opposed to full 3 3 Gauss,Legendre integration. This ,softens' these element, thereby increasing accuracy, albeit at the introduction of spurious zero energy modes on the element level. This is in general not considered problematic for the ,hourglass' mode common to Q8 and Q9 elements, since this spurious mode is non-communicable. The remaining two zero energy modes occurring in the Q9 element are indeed communicable. However, in topology optimization for instance, conditions may arise where the non-communicable spurious mode associated with the elements becomes activated. To effectively suppress these modes altogether in elements employing quadratic interpolation fields, two modified quadratures are employed herein. For the Q8 and Q9 membrane elements, the respective rules are a five and an eight point rule. As compared to fully integrated elements, the new rules enhance element accuracy due to the introduction of soft, higher-order deformation modes. A number of standard test problems reveal that element accuracy remains comparable to that of the under-integrated counterparts. Copyright 2004 John Wiley & Sons, Ltd. [source]


A geometry projection method for shape optimization

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 14 2004
J. Norato
Abstract We present a new method for shape optimization that uses an analytical description of the varying design geometry as the control in the optimization problem. A straightforward filtering technique projects the design geometry onto a fictitious analysis domain to support simplified response and sensitivity analysis. However, the analytical geometry model is referenced directly for all purely geometric calculations. The method thus combines the advantages of direct geometry representations with the simplified analysis procedures that are possible with fictitious domain analysis methods, such as the material distribution methods commonly used in topology optimization. The projected geometry measure converges to the indicator function of the analytical geometry model in the limit of numerical mesh refinement. Consequently, optimal designs obtained with the new method converge to solutions of well-defined continuum optimization problems in the limit of mesh refinement. This property is confirmed in example computations for minimum compliance design of an elastic structure subject to a volume constraint and for minimum volume design subject to a maximum stress constraint. Copyright 2004 John Wiley & Sons, Ltd. [source]


Designing materials with prescribed elastic properties using polygonal cells

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 3 2003
Alejandro R. Diaz
Abstract An extension of the material design problem is presented in which the base cell that characterizes the material microgeometry is polygonal. The setting is the familiar inverse homogenization problem as introduced by Sigmund. Using basic concepts in periodic planar tiling it is shown that base cells of very general geometries can be analysed within the standard topology optimization setting with little additional effort. In particular, the periodic homogenization problem defined on polygonal base cells that tile the plane can be replaced and analysed more efficiently by an equivalent problem that uses simple parallelograms as base cells. Different material layouts can be obtained by varying just two parameters that affect the geometry of the parallelogram, namely, the ratio of the lengths of the sides and the internal angle. This is an efficient way to organize the search of the design space for all possible single-scale material arrangements and could result in solutions that may be unreachable using a square or rectangular base cell. Examples illustrate the results. Copyright 2003 John Wiley & Sons, Ltd. [source]


Primal,dual Newton interior point methods in shape and topology optimization

NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 5-6 2004
R. H. W. Hoppe
Abstract We consider non-linear minimization problems with both equality and inequality constraints on the state variables and design parameters as they typically arise in shape and topology optimization. In particular, the state variables are subject to a partial differential equation or systems of partial differential equations describing the operating behaviour of the device or system to be optimized. For the numerical solution of the appropriately discretized problems we emphasize the use of all-in-one approaches where the numerical solution of the discretized state equations is an integral part of the optimization routine. Such an approach is given by primal,dual Newton interior point methods which we present combined with a suitable steplength selection and a watchdog strategy for convergence monitoring. As applications, we deal with the topology optimization of electric drives for high power electromotors and with the shape optimization of biotemplated microcellular biomorphic ceramics based on homogenization modelling. Copyright 2004 John Wiley & Sons, Ltd. [source]