Level Set Values Research Articles

Isoline Tracking in Particle-Based Fluids Using Level-Set Learning Representation

In this paper, we propose a learning model for tracking the isolines of fluid based on the physical properties of particles in particle-based fluid simulations. Our method involves analyzing which weights, closely related to surface tracking among the various physical properties of fluid particles, are significant. These weights are used as input values for the learning algorithm, enabling relatively accurate isoline tracking. In addition, compared to existing learning models such as linear regression, LSTM (long short-term memory), and learning representation (1-layer) models, our method obtained superior surface tracking results without accumulating errors. By using our proposed network structure to track the fluid surface, it learns and predicts values derived from existing fluid simulation algorithms, eliminating the need for computational processes for level-set values and enabling real-time surface tracking. As the scale of the simulation increases, our method significantly reduces the time and resources consumed compared to traditional methods and can track the fluid surface without additional resource consumption. Furthermore, due to our method’s simple network structure, the time consumed in the initial process of loading the model into memory is faster than models such as CNN and LSTM. Our proposed model occupies less than 30 kb of storage space, making it suitable for use in middleware. Lastly, to verify the generality of our method, we conducted tests in a total of five scenes, and in all test scenes, visually natural fluid isolines were represented.

Heat transfer efficiency enhancement of gyroid heat exchanger based on multidimensional gradient structure design

Nowadays, a bio-inspired heat exchanger incorporating a triply periodic minimal surface (TPMS) structure has demonstrated great potential in the fields of new energy research and aerospace, which necessitates achieving a balance between low volume and high heat transfer efficiency while maintaining low pressure drop. In this paper, the adjustable heat efficiency of the heat exchanger with TPMS in gradient thicknesses is specially designed. A steady-state conjugate heat transfer (CHT) model is coupled with computational fluid dynamics (CFD) analysis. In addition, the temperature profile and velocity streamline are also checked to analyze the fluid flow behavior of the radiator. The results show that the convective heat transfer coefficient of the Gyroid with gradient level set values is 26.02–60.10% higher than that of the uniform Gyroid model, and the pressure drop is decreased by 9.66–18.05%. Both high heat transfer efficiency and low pressure drop can be achieved when the thickness is 0.2–0.3 mm and Re is 100–125. The heat exchangers with a TPMS thickness gradient in the ratio of 2:4:6 demonstrate a remarkable enhancement in overall heat transfer efficiency, achieving a 30.22% improvement compared to those with a TPMS thickness gradient in 6:4:2.

Study on the topological morphology and mechanical properties of variable-amplitude TPMS structures

TPMS structures possess immense potential in terms of lightweight and multifunctional applications. However, previous studies have predominantly focused on aspects such as wall thickness, unit cell size, periodicity, and level set values. The impact of amplitude factors on the topological shape and mechanical properties of TPMS has not been fully elucidated. Here, primitive (P), gyroid (G), and diamond (D) structures were initially examined to explore the effects of amplitude factors on the topological shape and relative density of TPMS structures. The mechanical properties and energy absorption capability of the variable amplitude structures were investigated, with emphasis on a gyroid structure, using both experiments and numerical simulations. It was found that the amplitude does not alter the deformation pattern of the structure. However, the mechanical properties of the structure can be regulated over a relatively large range. Specifically, the elastic modulus and yield strength at an amplitude of 1.6 were observed to be approximately 1.5 times those at an amplitude of 0.4, and the plateau stress was approximately 1.33 times that of 0.4. Additionally, with increasing amplitude factor, there was a slight decrease in the densification strain of the structure but a significant increase in its energy absorption capability. Such findings offer a novel means by which the mechanical properties of TPMS structures can be modulated, providing new opportunities for the development of tissue scaffolds or energy-absorbing devices.

Improved XFEM for multiple crack analysis: Accurate and efficient implementations for stress intensity factors

The extended finite element method (XFEM) has achieved unprecedented success in crack analysis. However, challenges still remain for a multiple crack simulation. One issue is the difficulty of level set construction, where cracks are generally represented by combinations of different level set functions. Another issue is the rapidly increasing condition number of the global stiffness matrix, which is even more severe than the single crack case. In order to overcome these issues, we make two improvements as follows. On the one hand, inspired by the discontinuous description via cover cutting in Numerical Manifold Method (and later phantom node method in XFEM), we propose a level set templated cover cutting method, which makes use of level set values to cut a nodal patch and then to add virtual nodes. This method, which combines the advantages of both the level set method and the cover cutting technique, is simple and straightforward to implement. The method also plays a role in templated subdivision of discontinuous elements and hence presents an efficient and robust integration scheme. On the other hand, we extend the Improved XFEM (IXFEM), previously proposed by our research group, to the scenario of multiple crack problems. The method fundamentally eliminates the daunting issues of linear dependence and ill-conditioning of the standard XFEM, because it uses an extra-dof-free singularity enrichment around the crack tip. Numerical studies on multiple crack problems show that the developed approach offers various advantages: (1) Highly accurate SIF evaluation over the standard XFEM; (2) Well-conditioning of the global stiffness matrix independent of the number of cracks — condition number being of the same order as the standard FEM; (3) Efficient and robust linear system solving and geometric computations. Thus the developed approach is well capable of modeling arbitrary multiple crack problems.

Mechanical Behavior of Al-Si10-Mg P-TPMS Structure Fabricated by Selective Laser Melting and a Unified Mathematical Model with Geometrical Parameter.

Compared with the traditional lattice structure, the triply periodic minimal surface (TPMS) structure can avoid stress concentration effectively. Here, it is promising in the fields of lightweight and energy absorption. However, the number of structural parameters and mechanical properties of the TPMS structure is plentiful, and the relationship between them is unclassified. In this paper, for the first time, a unified mathematical model was proposed to establish the relationship between TPMS structural design parameters and mechanical properties. Fifteen primitive models were designed by changing the structural parameters (level-set value C and thickness T) and manufacturing by selective laser melting. The geometric defects and surface quality of the structures were explored by optical microscope and scanning electron microscopy (SEM). The mechanical properties were investigated by quasi-static compression test and finite element simulation. The influence of building direction on structural mechanical behavior (failure mode, stress-strain curve) was studied. The real mechanical properties (Young's modulus and plateau stress) of the structure could be predicted according to different C and T combinations. Finally, the energy absorption characteristics were explored. The results showed that when the C value is 0.6 in the range of 0-0.6, the energy absorption performance of the structure is at the maximum level.

DNS analysis of turbulent vaporizing two-phase flows, Part I: Topology of the velocity field

This work is devoted to the characterization of turbulence in two-phase flows with mass transfer. The objective is to provide further insights into the influence of liquid/gas interfaces on some statistical characteristics of the fluctuating velocity and scalar fields collected in the gas phase. The first part of this study is primarily focused on the velocity field. Thus, turbulence topology is educed from the consideration of the invariants of the Reynolds stresses, velocity gradients, strain-rate, and rotation tensors. This analysis is completed by some geometrical and alignment statistics. The corresponding quantities are studied using a set of direct numerical simulation (DNS) databases featuring two distinct values of the liquid volume fraction ϕℓ. The starting point is the transport equation for the mixture fraction. In most descriptions of two-phase flow combustion, it is indeed used to characterize the multi-component mixture. However, considering the non-linear dependence of chemical reaction rates to composition, the single knowledge of its mean (or filtered) state is insufficient to describe turbulent conditions. Its variance and associated mean (or filtered) scalar dissipation rate (SDR) must be evaluated, with the latter quantity (i.e., the SDR) driven, at leading order, by the velocity gradients relevant to straining effects and by the curvature of the mixture fraction field. The present statistical analysis makes use of conditional sampling based on a level-set value which signs the distance to the interface, thus allowing to discriminate its influence on the velocity gradients and associated turbulence topology.

Three dimensional vibroacoustic topology optimization of hearing instruments using cut elements

The focus of this article is on level set based topology optimization of vibroacoustic hearing instruments. The goal is to demonstrate the applicability of the proposed framework for optimization of 3D industrial scale hearing instruments. The framework employs an immersed boundary cut element method to handle the modeling of complex design geometries on fixed unstructured meshes. Utilization of unstructured meshes allows for optimizing small parts of a complex hearing instrument without disturbing the overall geometry. The remaining parts of the model which are not included in the optimization process are modeled with segregated approach. The design parameterization is based on an explicit level set approach, for which the nodal level set values are linked to the mathematical design variables. The optimization problem is solved using mathematical programming and the sensitivities are obtained with a discrete adjoint approach. A validation study is carried out comparing the proposed cut element model to a body fitted mesh for a large range of frequencies. The optimization framework is then applied on the tube and the suspension structures of a hearing instrument system considering two sets of material properties for the design parts. The optimization considers the minimization of sound pressure on the microphone surface for discrete frequencies aiming to reduce the feedback paths and to increase the amplification that the device can deliver to the user. Both optimization cases improve the performance of the hearing instrument system by effectively reducing the sound pressure on the microphone surface for the considered frequency range.

Level Set-Based Structural Optimization With Uniform Wall Thickness for Wire-Fed Metal Additive Manufacturing

Abstract A new three-dimensional structural optimization is presented based on the level set method to obtain favorable designs for wire-fed metal additive manufacturing with uniform wall thickness. By exploiting the signed distance nature of a level set function, a structure under design is always defined as a thin domain with uniform thickness without employing any constrains or penalty functionals. The boundary surfaces of a thin-walled domain are defined as the surfaces with level set values of ±t/2(t: wall thickness). Design velocity can be represented in terms of curvatures of the zero-level-set surface, extended to level set grids in the narrow band. Therefore, the calculation of accurate curvatures on the zero-level set is crucial for correct design sensitivities. In this investigation, mean and Gaussian curvatures at a point on the triangle mesh of the discretized zero-level set are calculated by spatial averages over the Voronoi cell of the point, by which the sensitivity of a material volume can be calculated with optimal accuracy. To address the high computational cost by a dense regular mesh for representing thin walls, degrees of freedom in void regions is mostly removed. Design examples of beams and a T-joint structure with uniform thickness are presented to verify the effectiveness of the proposed method.

Mechanical behavior of Al–Si10–Mg gyroid surface with variable topological parameters fabricated via laser powder bed fusion

Metallic cellular materials with triply periodic minimal surface (TPMS) topologies have great potential for lightweight and multi-functional applications. However, the effect of level-set values, which can control the topology of TPMS, on the performance of TPMS have not been fully understood because previous studies have mainly focused on the shell thickness, cell size and periodicity of TPMS. In this paper, Al–Si10–Mg gyroid surfaces with three different level-set values are fabricated by using selective laser melting (SLM). Micro computed tomography (Micro-CT) is used to reconstruct the 3D models of as-built samples, then the geometric deviations and surface quality are investigated. Mechanical behavior and energy absorption characteristics of these gyroid surfaces are investigated both experimentally and numerically, which provide a solid basis for predicting and controlling the deformation behavior in gyroid surfaces. Afterwards, the Gibson–Ashby model is used for understanding the effect of the level-set value on Young's modulus and plateau stress of gyroid surfaces. Finally, the relationship between level-set values and densification strain and energy absorbed per unit volume is also presented to gain insight into the role of level-set values on the energy absorption capability of the gyroid surfaces. Such understanding provides a new method to develop desired properties of implants or energy absorption applications by selecting an optimal combination of level-set value, thickness, periodicity etc.

XFEM Analyses Using Two-Dimensional Quadrilateral Elements Enriched with Only the Heaviside Step Function

In the framework of the extended finite element method, a two-dimensional four-node quadrilateral element enriched with only the Heaviside step function is formulated for stationary and propagating crack analyses. In the proposed method, two types of signed distance functions are used to implicitly express crack geometry, and finite elements, which interact with the crack, are appropriately partitioned according to the level set values and are then integrated numerically for derivation of the stiffness matrix and internal force vectors. The proposed method was verified by evaluating stress intensity factors, performing crack propagation analyses and comparing the obtained results with reference solutions.

Arbitrary 3D crack propagation with Improved XFEM: Accurate and efficient crack geometries

Extended finite element method (XFEM) achieves unprecedented success in crack growth simulations, in particular in 2D. However, challenges still remain for a 3D crack growth simulation. Among other issues like increased computational expense, one challenge is robust and efficient geometry representation of the non-planar surface and the usually curved front of the 3D crack, two unique features distinguished from its 2D counterpart. With a special emphasis on accurate geometries of the 3D crack, we propose in this paper a B-spline ruled surface (BRS), a smooth, explicit, meshfree crack representation algorithm, which offers good geometrical accuracy, good numerical efficiency, and no mesh-resolution dependence. The technique accurately represents the geometry of an evolving 3D crack in a lightweight manner and leads to highly efficient point-surface projections for level set values. The technique is applied to modeling arbitrary 3D crack propagation in context of the Improved XFEM (IXFEM) that has been developed to eliminate the daunting issues of linear dependence and ill-conditioning of the standard XFEM (Tian and Wen (2015); Wen and Tian (2016)). Numerical tests show that IXFEM with the technique of crack description provides accurate solutions for the benchmarks of 3D crack propagation; the effectiveness of the proposed algorithm is demonstrated.

A Deep Learning Approach for the Computation of Curvature in the Level-Set Method

We propose a deep learning strategy to estimate the mean curvature of two-dimensional implicit interfaces in the level-set method. Our approach is based on fitting feed-forward neural networks to synthetic data sets constructed from circular interfaces immersed in uniform grids of various resolutions. These multilayer perceptrons process the level-set values from mesh points next to the free boundary and output the dimensionless curvature at their closest locations on the interface. Accuracy analyses involving irregular interfaces, in both uniform and adaptive grids, show that our models are competitive with traditional numerical schemes in the $L^1$ and $L^2$ norms. In particular, our neural networks approximate curvature with comparable precision in coarse resolutions, when the interface features steep curvature regions, and when the number of iterations to reinitialize the level-set function is small. Although the conventional numerical approach is more robust than our framework, our results have unveiled the potential of machine learning for dealing with computational tasks where the level-set method is known to experience difficulties. We also establish that an application-dependent map of local resolutions to neural models can be devised to estimate mean curvature more effectively than a universal neural network.

Derivative-free level-set-based multi-objective topology optimization of flow channel designs using lattice Boltzmann method

A derivative-free level-set-based topology optimization method (DFLS-TO) was proposed and was used for channel structure design. Unlike in a conventional level-set method, quantum-behaved particle swarm optimization (QPSO) was used in DFLS-TO to optimize the level-set values at the knot points. The values at the non-knot points were interpolated by solving the Laplace equation. QPSO was combined with the Tchebycheff decomposition method to search for optimal level-set values in multi-objective problems. The channel boundary was represented by an iso-contour of the level-set function, and the B-spline method was used to convert the zigzag-like boundary into a smooth boundary. The lattice Boltzmann method (LBM) was integrated with the immersed boundary method to simulate the fluid flow. The Spalart–Allmaras model was incorporated with the LBM during the simulation of turbulent flows. To demonstrate the applicability of DFLS-TO, optimization of a pipe bend and a fluid distributor were performed.

Three-dimensional topology optimization of auxetic metamaterial using isogeometric analysis and model order reduction

We present an approach for designing material micro-structures by using isogeometric analysis and parameterized level set method. Design variables, which are level set values associated with control points, are updated from the optimizer and represent the geometry of the unit cell. The computational efficiency is further improved in each iteration by employing reduced order modeling when solving linear systems of the equilibrium equations. We construct a reduced basis by reusing computed solutions from previous optimization steps, and a much smaller linear system of equations is solved on the reduced basis. Two- and three-dimensional numerical results show the effectiveness of the topology optimization algorithm coupled with the reduced basis approach in designing metamaterials.

A Coarea Formulation for Grid-Based Evaluation of Volume Integrals

Abstract We present a new method for computing volume integrals based on data sampled on a regular Cartesian grid. We treat the case where the domain is defined implicitly by an inequality, and the input data include sampled values of the defining function and the integrand. The method employs Federer’s coarea formula (Federer, 1969, Geometric Measure Theory, Grundlehren der mathematischen Wissenschaften, Springer) to convert the volume integral to a one-dimensional quadrature over level set values where the integrand is an integral over a level set surface. Application of any standard quadrature method produces an approximation of the integral over the continuous range as a weighted sum of integrals over level sets corresponding to a discrete set of values. The integral over each level set is evaluated using the grid-based approach presented by Yurtoglu et al. (2018, “Treat All Integrals as Volume Integrals: A Unified, Parallel, Grid-Based Method for Evaluation of Volume, Surface, and Path Integrals on Implicitly Defined Domains,” J. Comput. Inf. Sci. Eng., 18, p. 3). The new coarea method fills a need for computing volume integrals whose integrand cannot be written in terms of a vector potential. We present examples with known results, specifically integration of polynomials over the unit sphere. We also present Saye’s (2015, “High-Order Quadrature Methods for Implicitly Defined Surfaces and Volumes in Hyperrectangles,” SIAM J. Sci. Comput., 37) example of integrating a logarithmic integrand over the intersection of a bounding box with an open domain implicitly defined by a trigonometric polynomial. For the final examples, the input data is a grid of mixture ratios from a direct numerical simulation of fluid mixing, and we demonstrate that the grid-based coarea method applies to computing volume integrals when no analytical form of the implicit defining function is given. The method is highly parallelizable, and the results presented are obtained using a parallel implementation capable of producing results at interactive rates.

Topology optimization of blank holders based on a kriging-interpolated level-set method

ABSTRACT Blank holders play an important role in non-isothermal stamping, and blank holder optimization is thus investigated in this article. A kriging-interpolated level-set method based on a parallel infilling strategy is proposed. First, a new method is used to discretize the design domain of a blank holder. The first-stage kriging model between the nodes and level-set values is developed, and a new blank holder structure can be determined according to a level-set threshold. The second-stage kriging model between the level sets and forming quality is then developed. The second-stage kriging model is updated based on a parallel infilling strategy. The best level-set values are obtained via an improved particle swarm optimization, and the optimal blank holder structure is determined. The proposed method is validated using a cross-shaped part. The results show that the optimized blank holder mass is reduced by 28%, and the forming quality is improved significantly.

A geometric multigrid accelerated incompressible flow simulation over complex geometries

Two algorithms for a fast Cartesian mesh-based flow simulation over complex geometries are presented. As a first category, a geometric multigrid (MG) method using direct restriction of the Heaviside function, which is 0 in the body and 1 in the fluid, is presented. This method is easy to implement, readily parallelizable, and can be applied to any irregular domain. The validity and optimal performance of the current MG method are demonstrated by solving an analytically defined Poisson problem on an irregular domain. Another part of this paper is to introduce a novel efficient method for computing the signed distance function (SDF), which is a typical level-set value to represent bodies on Cartesian meshes. In the Cartesian mesh-based simulation, only the SDF near the body interface is accurately required. Based on this fact, the number of operations for computing the SDF can be significantly reduced by focusing on interface cells. In this study, the adaptive mesh refinement (AMR) strategy is employed to effectively trace the interface cells. By using these two algorithms, an in-house Cartesian mesh-based incompressible flow solver is developed. To demonstrate the applicability of the current approaches, well-known benchmark problems in 2D and 3D are simulated. Finally, as the most challenging case, simulations for flow over an underwater robot, namely Crabster, are presented.

Adaptive Cartesian cut-cell sharp interface method (aC3SIM) for three-dimensional multi-phase flows

A sharp interface method has been developed for treating interfacial discontinuities in compressible multi-phase fluids on three-dimensional Cartesian cut-cell grids. The evolution of the interfacial discontinuities in the Cartesian grid is captured by the level-set method. The intersections between interfacial fronts and Cartesian grids are interpolated using level-set function values at the vertices of Cartesian grids. Triangular surfaces are then constructed on the interfacial fronts. A novel cell merge method is used for complex topological changes. Jump conditions across discontinuous interfaces are enforced by reconstruction of interfacial flow variables using a constrained least-squares method. The inviscid flux across internal faces of the same fluid is calculated by the local Lax–Friedrichs flux. Manufactured solutions for interfaces are suggested for validation of the reconstruction method. Laplace’s law test results show that the present method drastically reduces the parasite currents compared to conventional interface treatment methods. Bubble rise problems also show the validity and accuracy of the proposed sharp interface method for immiscible two-phase fluids.

Thermal buckling optimization of variable angle tow fibre composite plates with gap/overlap free design

Composite structures in a thermal environment may suffer from buckling failures due to thermally induced compressive forces, leading to concerns over their application in environmentally aggressive conditions. This issue is addressed by proposing a novel approach to the design of tow paths for enhancing thermal buckling performance, and it can also overcome the problems of tow gaps and overlapping. A level set function, such as defined by signed distance function, for representing a series of equidistant tows throughout the laminate, is adopted here to formulate an optimization problem that seeks to maximize the buckling load under thermal loading, and the level set values are thus the design variables. Sensitivities of thermal buckling eigenvalues with respect to tow paths, i.e. level set values, are derived through the adjoint method, and they are used to solve the optimization problem through the Hamilton-Jacobi equation. In this study, numerical examples are presented to demonstrate the effectiveness of the proposed method, where laminated plates made of various materials under different boundary conditions are considered. Results show that the proposed approach can provide efficient solutions with enhanced buckling performance.

Surface treatment technique of MPS method for free surface flows

A surface treatment technique is developed for moving particle semi-implicit (MPS) method in the simulation of incompressible free surface flows. The free surface particles are accurately detected by a developed geometrical method. Material interface is reconstructed at the interface of the fluid. According to the definition of level set function, the level set value of each particle is set to be the signed distance to the interface. The normal vector and curvature of interface are calculated at the position of interface via level set function, which help to obtain the accurate calculation of surface tension. The smoothness of level set function improves the computational accuracy of the interface curvature. The computational curvature has high convergence in current method. For example, the computational curvature of a circle has 1.75-order convergence. Benchmark tests including square droplet oscillation, head-on and off-center binary collision illustrate the advantages of current method. The developed method has higher accuracy and stability in the simulation of low speed incompressible free surface flows.