Level set topology optimization of cooling channels using the Darcy flow model

Structural and Multidisciplinary Optimization - The level set topology optimization method for 2D and 3D cooling channels, considering convective heat transfer for high Reynolds number flows, is...     

Level set based robust shape and topology optimization under random field uncertainties

A robust shape and topology optimization (RSTO) approach with consideration of random field uncertainty in loading and material properties is developed in this work. The proposed approach integrates the state-of-the-art level set methods for shape and topology optimization and the latest research development in design under uncertainty. To characterize the high-dimensional random-field uncertainty with a reduced set of random variables, the Karhunen–Loeve expansion is employed. The univariate dimension-reduction (UDR) method combined with Gauss-type quadrature sampling is then employed for calculating statistical moments of the design response. The combination of the above techniques greatly reduces the computational cost in evaluating the statistical moments and enables a semi-analytical approach that evaluates the shape sensitivity of the statistical moments using shape sensitivity at each quadrature node. The applications of our approach to structure and compliant mechanism designs show that the proposed RSTO method can lead to designs with completely different topologies and superior robustness.     

Level set topology optimization of cooling and heating devices using a simplified convection model

This paper studies topology optimization of convective heat transfer problems in two and three dimensions. The convective fluxes are approximated by Newton’s Law of Cooling (NLC). The geometry is described by a Level Set Method (LSM) and the temperature field is predicted by the eXtended Finite Element Method (XFEM). A constraint on the spatial gradient of the level set field is introduced to penalize small, sub-element-size geometric features. Numerical studies show that the LSM-XFEM provides improved accuracy over previously studied density methods and LSMs using Ersatz material models. It is shown that the NLC model with an iso-thermal fluid phase may over predict the convective heat flux and thus promote the formation of very thin fluid channels, depending on the Biot number characterizing the heat transfer problem. Approximating the temperature field in the fluid phase by a diffusive model mitigates this issue but an explicit feature size control is still necessary to prevent the formation of small solid members, in particular at low Biot numbers. The proposed constraint on the gradient of the level set field is shown to suppress sub-element-size features but necessitates a continuation strategy to prevent the optimization process from stagnating as geometric features merge.     

Stress-related topology optimization via level set approach

Considering stress-related objective or constraint functions in structural topology optimization problems is very important from both theoretical and application perspectives. It has been known, however, that stress-related topology optimization problem is challenging since several difficulties must be overcome in order to solve it effectively. Traditionally, SIMP (Solid Isotropic Material with Penalization) method was often employed to tackle it. Although some remarkable achievements have been made with this computational framework, there are still some issues requiring further explorations. In the present work, stress-related topology optimization problems are investigated via a level set-based approach, which is a different topology optimization framework from SIMP. Numerical examples show that under appropriate problem formulations, level set approach is a promising tool for stress-related topology optimization problems.     

Evolutionary level set method for structural topology optimization

This paper proposes an evolutionary accelerated computational level set algorithm for structure topology optimization. It integrates the merits of evolutionary structure optimization (ESO) and level set method (LSM). Traditional LSM algorithm is largely dependent on the initial guess topology. The proposed method combines the merits of ESO techniques with those of LSM algorithm, while allowing new holes to be automatically generated in low strain energy within the nodal neighboring region during optimization. The validity and robustness of the new algorithm are supported by some widely used benchmark examples in topology optimization. Numerical computations show that optimization convergence is accelerated effectively.     

Level set topology and shape optimization by density methods using cut elements with length scale control

Structural and Multidisciplinary Optimization - The level set and density methods for topology optimization are often perceived as two very different approaches. This has to some extent led to two...     

Application of spectral level set methodology in topology optimization

In this paper, two benchmark problems in structural boundary design are solved using the spectral level set methodology, which is a new approach to topology optimization of interfaces. This methodology is an extension of the level set methods, in which the interface is represented as the zero level set of a function. According to the proposed formulation, the Fourier coefficients of that function are the design variables describing the interface during the topology optimization. An advantage of the spectral level set methodology, in the case of a sufficiently regular interface, is to admit an upper bound error which is asymptotically smaller than the one for nonadaptive spacial discretizations of the level set function. Other advantages include the nucleation of holes in the interior of the interface and the avoidance of checkerboard-like designs. The theoretical framework of the methodology is presented and estimates on its convergence rate are discussed. The numerical applications consist in the design of short and long cantilevers subject to a vertical concentrated load opposite to the fixed end. The goal is to maximize the structural stiffness subject to a solid volume constraint. The zero level set of the level set function defines the structural boundary.     

A subtractive manufacturing constraint for level set topology optimization

Structural and Multidisciplinary Optimization - We present a method for enforcing manufacturability constraints in generated parts such that they will be automatically ready for fabrication using a...     

Stress and strain control via level set topology optimization

This paper presents a level set topology optimization method for manipulation of stress and strain integral functions in a prescribed region (herein called sub-structure) of a linear elastic domain. The method is able to deviate or concentrate the flux of stress in the sub-structure by optimizing the shape and topologies of the boundaries outside of that region. A general integral objective function is proposed and its shape sensitivities are derived. For stress isolation or maximization, a von Mises stress integral is used and results show that stresses in the sub-structure can be drastically reduced. For strain control, a strain integral combined with a vector able to select the component of the strain is used. A combination of both can be used to minimize deformation of a prescribed direction. Numerical results show that strain can be efficiently minimized or maximized for a wide range of directions. The proposed methodology can be applied to stress isolation of highly sensitive non strain-based sensors, design for failure, maximization of mechanical strain and strain direction control for strain-based sensors and microdevices.     

Single-loop system reliability-based topology optimization considering statistical dependence between limit-states

This paper presents a single-loop algorithm for system reliability-based topology optimization (SRBTO) that can account for statistical dependence between multiple limit-states, and its applications to computationally demanding topology optimization (TO) problems. A single-loop reliability-based design optimization (RBDO) algorithm replaces the inner-loop iterations to evaluate probabilistic constraints by a non-iterative approximation. The proposed single-loop SRBTO algorithm accounts for the statistical dependence between the limit-states by using the matrix-based system reliability (MSR) method to compute the system failure probability and its parameter sensitivities. The SRBTO/MSR approach is applicable to general system events including series, parallel, cut-set and link-set systems and provides the gradients of the system failure probability to facilitate gradient-based optimization. In most RBTO applications, probabilistic constraints are evaluated by use of the first-order reliability method for efficiency. In order to improve the accuracy of the reliability calculations for RBDO or RBTO problems with high nonlinearity, we introduce a new single-loop RBDO scheme utilizing the second-order reliability method and implement it to the proposed SRBTO algorithm. Moreover, in order to overcome challenges in applying the proposed algorithm to computationally demanding topology optimization problems, we utilize the multiresolution topology optimization (MTOP) method, which achieves computational efficiency in topology optimization by assigning different levels of resolutions to three meshes representing finite element analysis, design variables and material density distribution respectively. The paper provides numerical examples of two- and three-dimensional topology optimization problems to demonstrate the proposed SRBTO algorithm and its applications. The optimal topologies from deterministic, component and system RBTOs are compared with one another to investigate the impact of optimization schemes on final topologies. Monte Carlo simulations are also performed to verify the accuracy of the failure probabilities computed by the proposed approach.     

Structural optimization based on topology optimization techniques using frame elements considering cross-sectional properties

This paper discusses a new structural optimization method, based on topology optimization techniques, using frame elements where the cross-sectional properties can be treated as design variables. For each of the frame elements, the rotational angle denoting the principal direction of the second moment of inertia is included as a design variable, and a procedure to obtain the optimal angle is derived from Karush–Kuhn–Tucker (KKT) conditions and a complementary strain energy-based approach. Based on the above, the optimal rotational angle of each frame element is obtained as a function of the balance of the internal moments. The above methodologies are applied to problems of minimizing the mean compliance and maximizing the eigen frequencies. Several examples are provided to show the utility of the presented methodology.     

Large-scale level set topology optimization for elasticity and heat conduction

Structural and Multidisciplinary Optimization - This paper addresses the issue of minimizing support material in additive manufacturing (AM) during topology optimization (TO) in order to reduce...     

Hole seeding in level set topology optimization via density fields

Structural and Multidisciplinary Optimization - Two approaches that use a density field for seeding holes in level set topology optimization are proposed. In these approaches, the level set field...     

Design of pipeline opening layout through level set topology optimization

Perforated pipeline structure is widely utilized in the oil industry for its special functionality of communicating media with the ambient environment. A typical application is the slotted liner in SAGD (Steam Assisted Gravity Drainage) process, where the pipeline structure is manufactured with open slots to spread hot steam and collect the melted oil. Generally, a dense opening layout is employed to reduce flow resistance. On the other hand, inclusion of the many openings severely reduces the structural strength and stiffness, which causes the pipeline prone to deformation or even failure. Therefore, there exist the two conflicting requirements for design of the pipeline opening layout, and an interesting solution is proposed in this paper. To be specific, the pipeline structure is discretized into shell elements which are categorized into multiple types: without opening, with opening type 1, with opening type 2, etc. These element types are treated as different material phases, and design of the pipeline opening layout is transformed into a multi-material topology optimization problem. Multi-material level set method is employed to solve it, subject to the compliance minimization objective. In addition, a lower bound of opening quantity is applied by properly configuring the material fraction constraint, which ensures the low flow resistance. The effectiveness of the proposed method is proven through a few numerical case studies.     

Eigenvalue topology optimization of structures using a parameterized level set method

Preventing a structure from resonance is important in many real-world applications. Because an external excitation frequency can be lower than the fundamental eigenfrequency or between the eigenfrequencies of a structure, there is a strong need for eigenfrequency optimization technology to optimize the fundamental eigenfrequency and, in addition, the k-th eigenfrequency and to maximize the gap between eigenfrequencies. However, previous optimization studies on vibrating elastic structures that used the level set method have been devoted to the optimization of the fundamental eigenfrequency, whereas the higher-order eigenfrequencies optimization problem has seldom been considered. This paper presents an eigenfrequency optimization technology that is based on the compactly supported radial basis functions (CS-RBFs) parameterized level-set method, using the fundamental eigenfrequency, the eigenfrequency of a given higher-order, and the gap between two consecutive eigenfrequencies as the optimization objectives. Furthermore, to address the oscillation problem of the objective function, we adopt an exponential weighted optimization model of a number of the lower eigenfrequencies for multiple eigenvalue optimizations, and we utilize mode-tracking technology for the single eigenvalue optimization.In addition, we further extend the CS-RBFs parameterized level-set method to an optimization that is performed with geometric constraints, which means that the size and position of the regular holes in the structure can be optimized with the shape and topology. This approach is useful in real-world applications. The effectiveness of this method is demonstrated by several widely investigated examples that have various objectives.     

Shape and topology optimization based on the convected level set method

The aim of this research is to construct a shape optimization method based on the convected level set method, in which the level set function is defined as a truncated smooth function obtained by using a sinus filter based on a hyperbolic tangent function. The local property of the hyperbolic tangent function dramatically reduces the generation of red the error between the specified profile of the hyperbolic tangent function and the level set function that is updated using a time evolution equation. In addition, the small size of the error facilitates the use of convective reinitialization, whose basic idea is that the reinitialization is embedded in the time evolution equation, whereas such treatment is typically conducted in a separate calculation in conventional level set methods. The convected level set method can completely avoid the need for additional calculations when performing reinitialization. The validity and effectiveness of our presented method are tested with a mean compliance minimization problem and a problem for the design of a compliant mechanism.