Geometry Projection Method Research Articles

A geometry projection method for designing and optimizing additively manufactured variable-stiffness composite laminates

A method for designing laminates is presented using geometry projection to optimize the layout of additively manufactured variable-stiffness composite laminates. By considering fiberreinforced bars as geometric primitives, the geometry projection methodology is extended to include optimizing regions with intersecting load paths. This is achieved by utilizing a dual representation of bars, which considers the geometric parameters and the element-wise density field representation. The dual representation enables the combining and overlapping of bars, resulting in a localized orthotropic material response at overlapping regions that mitigates the transverse compliant response of fiberreinforced components. The proposed method’s effectiveness is demonstrated through minimizing the compliance of the Messerschmitt-Bölkow-Blohm beam problem, a well-known benchmark problem in topology optimization.

Topology optimization of magnetic cores for WPT using the geometry projection method

PurposeThe purpose of this study is to search for an optimal core shape that is robust against misalignment between the transmitting and receiving coils of the wireless power transfer (WPT) device. During the optimization process, the authors maximize the coupling coefficients while minimizing the leakage flux around the coils to ensure the safety of the WPT device.Design/methodology/approachIn this study, a novel topology optimization method for WPT devices using the geometry projection method is proposed to optimize the magnetic core shape. This method facilitates the generation of bar-shaped magnetic cores because the material distribution is represented by a set of elementary bars.FindingsIt is shown that an optimized core shape, which is obtained through topology optimization, effectively increases the net magnetic flux interlinked with the receiving coil and outperforms the conventional core.Originality/valueIn the previous topology optimization method, the material distribution is represented by a linear combination of Gaussian functions. However, this method does not usually result in bar-shaped cores, which are widely used in WPT. In this study, the authors propose a novel topology optimization method for WPT devices using geometry projection that is used in structural optimization, such as beam and cantilever shapes.

Topology optimization with discrete geometric components made of composite materials

We present a novel topology optimization (TO) method for the design of structures composed of bars that are made of an orthotropic, fiber-reinforced material. The designs generated under this framework tailor the layout of the fiber-reinforced bars to maximize the stiffness of the structure for a fixed amount of material. The proposed method extends the geometry projection method to represent the design via cylindrical bars that may be reinforced with continuous fibers and perform the analysis using a fixed finite element mesh. The interpolation of the material properties at the junction of multiple bars ensures that, out of the materials of the intersecting bars, the optimizer chooses the single best material. Numerical examples for the stiffness-based design of 2- and 3-dimensional frame structures are presented to demonstrate the efficacy of the proposed method.

Adaptive mesh refinement for topology optimization with discrete geometric components

This work introduces an Adaptive Mesh Refinement (AMR) strategy for the topology optimization of structures made of discrete geometric components using the geometry projection method. Practical structures made of geometric shapes such as bars and plates typically exhibit low volume fractions with respect to the volume of the design region they occupy. To maintain an accurate analysis and to ensure well-defined sensitivities in the geometry projection, it is required that the element size is smaller than the smallest dimension of each component. For low-volume-fraction structures, this leads to finite element meshes with very large numbers of elements. To improve the efficiency of the analysis and optimization, we propose a strategy to adaptively refine the mesh and reduce the number of elements by having a finer mesh on the geometric components, and a coarser mesh away from them. The refinement indicator stems very naturally from the geometry projection and is thus straightforward to implement. We demonstrate the effectiveness of the proposed AMR method by performing topology optimization for the design of minimum-compliance and stress-constrained structures made of bars and plates.

Multi-material topology optimization of lattice structures using geometry projection

This work presents a computational method for the design of architected truss lattice materials where each strut can be made of one of a set of available materials. We design the lattices to extremize effective properties. As customary in topology optimization, we design a periodic unit cell of the lattice and obtain the effective properties via numerical homogenization. Each bar is represented as a cylindrical offset surface of a medial axis parameterized by the positions of the endpoints of the medial axis. These parameters are smoothly mapped onto a continuous density field for the primal and sensitivity analysis via the geometry projection method. A size variable per material is ascribed to each bar and penalized as in density-based topology optimization to facilitate the entire removal of bars from the design. During the optimization, we allow bars to be made of a mixture of the available materials. However, to ensure each bar is either exclusively made of one material or removed altogether from the optimal design, we impose optimization constraints that ensure each size variable is 0 or 1, and that at most one material size variable is 1. The proposed material interpolation scheme readily accommodates any number of materials. To obtain lattices with desired material symmetries, we design only a reference region of the unit cell and reflect its geometry projection with respect to the appropriate planes of symmetry. Also, to ensure bars remain whole upon reflection inside the unit cell or with respect to the periodic boundaries, we impose a no-cut constraint on the bars. We demonstrate the efficacy of our method via numerical examples of bulk and shear moduli maximization and Poisson’s ratio minimization for two- and three-material lattices with cubic symmetry.

Generalized Geometry Projection: A Unified Approach for Geometric Feature Based Topology Optimization

Structural topology optimization has seen many methodological advances in the past few decades. In this work we focus on continuum-based structural topology optimization and more specifically on geomet- ric feature based approaches, also known as explicit topology optimization, in which a design is described as the assembly of simple geometric components that can change position, size and orientation in the con- sidered design space. We first review various recent de- velopments in explicit topology optimization. We then describe in details three of the reviewed frameworks, which are the Geometry Projection method, the Moving Morphable Components with Esartz material method

Topology Optimization of Structures Made of Discrete Geometric Components With Different Materials

We present a new method for the simultaneous topology optimization and material selection of structures made by the union of discrete geometric components, where each component is made of one of multiple available materials. Our approach is based on the geometry projection method, whereby an analytical description of the geometric components is smoothly mapped onto a density field on a fixed analysis grid. In addition to the parameters that dictate the dimensions, position, and orientation of the component, a size variable per available material is ascribed to each component. A size variable value of unity indicates that the component is made of the corresponding material. Moreover, all size variables can be zero, signifying the component is entirely removed from the design. We penalize intermediate values of the size variables via an aggregate constraint in the optimization. We also introduce a mutual material exclusion constraint that ensures that at most one material has a unity size variable in each geometric component. In addition to these constraints, we propose a novel aggregation scheme to perform the union of geometric components with dissimilar materials. These ingredients facilitate treatment of the multi-material case. Our formulation can be readily extended to any number of materials. We demonstrate our method with several numerical examples.

A geometry projection method for the topology optimization of curved plate structures with placement bounds

SummarySingle‐curvature plates are commonly encountered in mechanical and civil structures. In this paper, we introduce a topology optimization method for the stiffness‐based design of structures made of curved plates with fixed thickness. The geometry of each curved plate is analytically and explicitly represented by its location, orientation, dimension, and curvature radius, and therefore, our method renders designs that are distinctly made of curved plates. To perform the primal and sensitivity analyses, we use the geometry projection method, which smoothly maps the analytical geometry of the curved plates onto a continuous density field defined over a fixed uniform finite element grid. A size variable is ascribed to each plate and penalized in the spirit of solid isotropic material with penalization, which allows the optimizer to remove a plate from the design. We also introduce in our method a constraint that ensures that no portion of a plate lies outside the design envelope. This prevents designs that would otherwise require cuts to the plates that may be very difficult to manufacture. We present numerical examples to demonstrate the validity and applicability of the proposed method.

Design and Additive Manufacturing of 3D Phononic Band Gap Structures Based on Gradient Based Optimization.

We present a novel approach for gradient based maximization of phononic band gaps. The approach is a geometry projection method combining parametric shape optimization with density based topology optimization. By this approach, we obtain, in a two dimension setting, cellular structures exhibiting relative and normalized band gaps of more than 8 and 1.6, respectively. The controlling parameter is the minimal strut size, which also corresponds with the obtained stiffness of the structure. The resulting design principle is manually interpreted into a three dimensional structure from which cellular metal samples are fabricated by selective electron beam melting. Frequency response diagrams experimentally verify the numerically determined phononic band gaps of the structures. The resulting structures have band gaps down to the audible frequency range, qualifying the structures for an application in noise isolation.

Stress-based topology optimization with discrete geometric components

In this paper, we introduce a framework for the stress-based topology optimization of structures made by the assembly of discrete geometric components, such as bars and plates, that are described by explicit geometry representations. To circumvent re-meshing upon design changes, we employ the geometry projection method to smoothly map the geometric components onto a continuous density field defined over a uniform finite element grid for analysis. The geometry projection is defined in a manner that prevents the singular optima phenomenon widely reported in the literature, and that effectively considers stresses only on the geometric components and not on the void region. As in previous work, a size variable is ascribed to each geometry component and penalized in the spirit of solid isotropic material with penalization (SIMP), allowing the optimizer to entirely remove geometric components from the design. We demonstrate our method on the L-bracket benchmark for stress-based optimization problems in 2-d and 3-d.

Realization of compact broadband optical diode in linear air-hole photonic crystal waveguide.

In this paper, we present a compact broadband design for reciprocal optical diode in linear two-dimensional air-hole photonic crystal waveguide. The forward even-to-odd mode conversion and backward blockade of even mode are achieved by introducing the functional region with 1.2a×2.8a area. The inside dielectric distribution is obtained by finite element method combining geometry projection method and the method of moving asymptotes. In our design, only one asymmetrically deformed air hole locates in the functional region. The parabola-like unidirectionality keeps higher than 15dB within the operational bandwidth 0.01c/a (about 40nm when 1550nm is the center wavelength), and the maximum value reaches approximate 24 dB near the center frequency. Meanwhile, the forward transmission efficiency keeps higher than 89.9%. Moreover, the optical diode effect of the proposed design is validated in three-dimensional model and the tolerance of the imperfection in fabricating is demonstrated as well. This compact broadband optical diode can contribute to the all-optical integrated circuits.

A geometry projection method for the topology optimization of plate structures

We introduce a topology optimization method for the stiffness-based design of structures made of plates. Our method renders topologies made distinctly of plates, thereby producing designs that better conform to manufacturing processes tailored to plate structures, such as those that employ stock plates that are cut and joined by various means. To force the structural members to be plates, we employ the geometry projection method to project an analytical description of a set of fixed-thickness plates onto a continuous density field defined over a 3-dimensional, uniform finite element grid for analysis. A size variable is assigned to each plate and penalized so that the optimizer can entirely remove a plate from the design. The proposed method accommodates the case where the plates in the topology are rectangular and solid, and the case where the boundaries of the plates can change and holes can be introduced. The latter case is attained by composition with a free density field. We present examples that demonstrate the effectiveness of our method and discuss future work.

A geometry projection method for continuum-based topology optimization with discrete elements

Abstract This article describes a method for the continuum-based topology optimization of structures made of discrete elements. In particular, we examine the optimization of linearly elastic planar structures made of bars of fixed width and semicircular ends. The design space for the optimization consists of the endpoint locations of the bar’s medial axes and their out-of-plane thicknesses. To circumvent re-meshing upon design changes, we project the design onto a fixed analysis grid using a differentiable geometry projection that results in a density field indicating the fraction of solid material anywhere in the design space, as in density-based topology optimization methods. The out-of-plane thickness is penalized so that the optimizer is capable of removing bars during the optimization. The differentiability of the projection allows for the computation via the chain rule of design sensitivities of responses of interest, and therefore it allows for the use of robust and efficient gradient-based optimization methods. Notably, this approach makes it easier to fabricate optimal designs by using off-the-shelf stock material. Furthermore, the method considers the case where bars overlap at their joints (i.e. their thicknesses are added at the joint) and when they do not. Finally, our proposed method naturally accommodates the imposition of several fixed length scales. We demonstrate the proposed approach on classical problems of compliance-based topology optimization and identify its benefits as well as research directions to be addressed in the future.

Ultra-compact broadband mode converter and optical diode based on linear rod-type photonic crystal waveguide.

In this paper, we present extremely compact designs of both broadband mode converter and optical diode in linear rod-type photonic crystal (PhC) waveguide with functional region consisting of only 4 × 1 unit cells of perfect PhC. The dielectric distribution inside functional region are optimized by combining geometry projection method and method of moving asymptotes. Bidirectional mode converter realizes above 60% transmission efficiency within bandwidth 0.02c/a, where c and a represent light velocity and PhC lattice constant respectively. Optical diode achieves above 19 dB unidirectionality for even mode within bandwidth 0.01c/a. Moreover, the proposed designs have reasonable tolerance of rod boundary fluctuation. We expect the results will help developing recipes for future PhC devices in all-optical integrated circuits.

Ultrasmall modal volume and high Q factor optimization of a photonic crystal slab cavity

We investigate the photonic crystal slab cavities with different symmetries (C2,σv, C3,σv, C6,σv), and try to find out the rules for increasing the Q factor and reducing the effective modal volume value. The results indicate that the cavity with C2,σv symmetry may be the most suitable candidate, which has high Q factor and small effective modal volume. Based on the cavity with C2,σv symmetry, we use the improved geometry projection method to optimize the shapes of neighbor holes of the cavity and obtain a new slab cavity structure with a high Q value of 5.15 × 105 and small effective modal volume value of 0.09(λ0/n)3. The performance of such a structure is better than in previous reports, which is significant for developing photonic crystal cavity based devices.

Optimization of a two-dimensional photonic crystal waveguide for ultraslow light propagation

A two-dimensional photonic crystal waveguide is optimized for ultraslow light propagation. By employing geometry projection method optimization, a flat band photonic crystal waveguide is achieved with the minimum group index around 2335, which is much larger than those reported before. The investigation of the transient propagation of the Gaussian pulse in the waveguide indicated that low group velocity dispersion can cause a small expansion of the pulse. The stability analysis indicates that slightly numerical and angular deviation in fabrication can influence the minimum group index of the waveguide, but the ultraslow light property can still be maintained. The theory simulation in our work can provide some useful guidance for further ultraslow light research.

Optimizing the top profile of a nanowire for maximum forward emission

The optimal top structure of a nanowire quantum emitter single photon source is significant in improving performance. Based on the axial symmetry of a cylindrical nanowire, this paper optimizes the top profile of a nanowire for the maximum forward emission by combining the geometry projection method and the finite element method. The results indicate that the nanowire with a cambered top has the stronger emission in the forward direction, which is helpful to improve the photon collection efficiency.

The optimal structure of two dimensional photonic crystals with the large absolute band gap

This paper reports a new designed square lattice GaAs structure of two-dimensional photonic crystals with absolute band gap approach to 0.1623 (2πc/a), where a is the period of the square lattice. The optimal structure is obtained by combining the Geometry Projection Method and Finite Element Method. Both gradient information and symmetric control points are introduced to reduce the calculation cost. For benefit to the fabrication in reality, the structure is simplified by the combination of triangle and rectangular geometry. Through parameter optimization, the absolute band gap of the new structure is improved to 0.1735 (2πc/a), which is much larger than those reported before. The new PC structure is convenient and stab for fabrication, and may be found applications in the future optical devices.

A Computational and Cellular Solids Approach to the Stiffness-Based Design of Bone Scaffolds

We derive a cellular solids approach to the design of bone scaffolds for stiffness and pore size. Specifically, we focus on scaffolds made of stacked, alternating, orthogonal layers of hydroxyapatite rods, such as those obtained via micro-robotic deposition, and aim to determine the rod diameter, spacing and overlap required to obtain specified elastic moduli and pore size. To validate and calibrate the cellular solids model, we employ a finite element model and determine the effective scaffold moduli via numerical homogenization. In order to perform an efficient, automated execution of the numerical studies, we employ a geometry projection method so that analyses corresponding to different scaffold dimensions can be performed on a fixed, non-conforming mesh. Based on the developed model, we provide design charts to aid in the selection of rod diameter, spacing and overlap to be used in the robotic deposition to attain desired elastic moduli and pore size.

Optimization of photonic nanostructures

A Geometry Projection Method (GPM) for the optimization of photonic crystal nanostructures is used to improve the performance of two different photonic devices. The GPM is a level set based approach for defining a differentiable expression describing the material distribution of a structure. It gives direct control over areas of intermediate dielectric constant and implicit control over feature size, both key concerns for the design of nano-scale devices. The GPM, in conjunction with the finite element and adjoint methods, is used here to solve two different non-linear optimization problems. The diffraction from a two-dimensional photonic crystal waveguide termination is reduced, and the reflectivity of a three-dimensional photonic crystal laser cavity structure is increased. Significant improvements in both cases demonstrate that the GPM is an effective tool for optimization of photonic structures.