Level Set Equation Research Articles

Multi-objective topology optimization and numerical investigation of heat sinks based on triply periodic minimal surface lattices

In thermal management applications, such as heat sinks (HSs) for electronic devices, cellular materials have extensively been employed. In recent years, there has been a growing attention towards employing topology optimization for enhancing hydraulic and heat transfer performance of HSs by optimizing their topology. The utilization of triply periodic minimal surface (TPMS) based structures presents distinctive prospects for customizing the design and performance of HSs. However, their potential remains unexplored in the context of customizing additively manufactured porous optimized HSs. Consequently, there is a need for research aimed at examining their coupled hydraulic and thermal performance. Density mapping approaches are used to build a variable density TPMS-based HSs from the output of topology optimization by applying the TPMS level-set equations that relate relative density and the level-set constant. In this work, a relative density mapping methodology is applied to thermo-fluid optimization problem to design a TPMS-based convective cooling system. An in-house MATLAB code was developed to perform a multi-objective topology optimization. After that, uniform and variable density (mapped from topology optimization results) TPMS-based HSs are analyzed using Star-CCM + CFD software to investigate their hydraulic and heat transfer performance. An experimental setup was established, and the numerical results were validated using uniform TPMS-based heat sinks. Results showed that incorporating TPMS with topology optimization has a great potential in thermal management applications as pressure drop across the heat sink was reduced while maintaining the performance.

Mechanism of temporal interface evolution and internal circulations during the droplet formation in a planar slit T-microchannel

The present study has numerically explored the mechanism of interface evolution and internal flow circulations during the droplet formation in two-phase flow through a planar T-microchannel. The two-dimensional unsteady form of the conservative level set equation coupled with Navier–Stokes equations has been solved using the finite element method. The range of parameters include the contact angle (θ) from 120° to 180°, and the flow rate ratio (Qr) from 0.1 to 10 for the low capillary number (Cac≤10−2). The present study indicates that surface wettability plays a crucial role in influencing the temporal evolution of the interface. The internal flow circulation in the droplet is controlled by the axial and radial velocities primarily influenced by shear stress. The newly introduced novel “interface-to-neck ratio” parameter has provided another platform to investigate the pinch-off dynamics of droplets. Moreover, the phenomenon of droplet pinch-off is primarily initiated and driven by the Laplace pressure, defined by three distinct approaches: the pressure difference method, the determination of the minimum local radius of curvature on the rear side, and a calculation of the neck width. The predictive correlations have been established to estimate the droplet characteristics as a function of the flow rate ratio and contact angle. The findings reported have significant implications for the design of droplet dispensing systems that depend on surface wettability as a critical regulating parameter.

Large-eddy simulation study on wave-wake interactions past an offshore wind turbine

Large-eddy simulations (LES) of the flow past an offshore wind turbine under different underlying wave fields have been performed. A one-way coupling between the air and water flows has been realized through a hybrid immersed-boundary/level-set method. The velocity in the water domain is forced with the potential flow solution, while the response of the atmospheric boundary layer to the changes in the sea-surface height (tracked by the level-set equation) is simulated with a LES approach coupled to a rotating actuator disk model to mimic the effect of the wind turbine. A parametric study has been performed varying the wave period and wavelength while keeping the amplitude constant, resulting in different wave age parameters ranging from young developing waves to old swell waves. The wave field has a significant effect on the lower region of the atmospheric boundary layer, slowing down the wind field in proximity of the air-water interface and considerably increasing the local turbulence kinetic energy (TKE). The interaction between the wave evolution and the TKE results in a non-monotonic trend of the wake recovery rate with the wave age, when compared to the baseline value in the wake of a turbine over a flat wall. Both developed waves results in a lower recovery rate, whereas intermediate-age waves present a larger value than the baseline. The increased TKE in the lower layers of the rotor revolution induces an increased fluctuating component in the power production and blade loads. Nevertheless, for the parameters considered in this study, the spectra of the blade loads do not show a clear signature at the wave frequency, but the increased fluctuating component occurs over a broad range of frequencies.

Mathematical analysis of modified level-set equations

The linear transport equation allows to advect level-set functions to represent moving sharp interfaces in multiphase flows as zero level-sets. A recent development in computational fluid dynamics is to modify the linear transport equation by introducing a nonlinear term to preserve certain geometrical features of the level-set function, where the zero level-set must stay invariant under the modification. The present work establishes mathematical justification for a specific class of modified level-set equations on a bounded domain, generated by a given smooth velocity field in the framework of the initial/boundary value problem of Hamilton–Jacobi equations. The first main result is the existence of smooth solutions defined in a time-global tubular neighborhood of the zero level-set, where an infinite iteration of the method of characteristics within a fixed small time interval is demonstrated; the smooth solution is shown to possess the desired geometrical feature. The second main result is the existence of time-global viscosity solutions defined in the whole domain, where standard Perron’s method and the comparison principle are exploited. In the first and second main results, the zero level-set is shown to be identical with the original one. The third main result is that the viscosity solution coincides with the local-in-space smooth solution in a time-global tubular neighborhood of the zero level-set, where a new aspect of localized doubling the number of variables is utilized.

On development of lattice-Boltzmann method based multi phase flow solver

A numerical methodology based on lattice Boltzmann and level set function is presented to simulate the two-phase flows with stringent density and viscosity ratio. Two different strategies - single grid (SGLBLS) and dual grid (DGLBLS) are used. For accurate computation of interfacial forces in DGLBLS, the level set equations are solved on a uniform grid which is twice that for lattice-Boltzmann equations. The code validation and performance of SGLBLS and DGLBLS is carried on three different problems: Young’s Laplace law, 2D rising bubble and Rayleigh–Taylor instability. The results show that the representation of the interface is more accurate in DGLBLS with significantly less mass error - establishing it as an accurate method to handle stringent density and viscosity ratio. The comparison between DGLBLS and SGLBLS shows that the DGLBLS leads to a large improvement in the accuracy of the results with only slight increase in the computational cost.

Hydrodynamic expansion and plume splitting of the ultrafast laser-induced plasma during ablation of multi-element metallic materials under atmospheric condition

In this study, the hydrodynamic expansion and splitting effect of ultrafast laser-induced plasma plume for multi-element alloys were studied. A fully coupled hydrodynamic model for femtosecond laser ablation of multi-element alloys was presented to study the ambipolar electric field during the plume expansion process. The model utilized a level-set equation to capture the interface between the condensed phase and the gaseous phase and accounted for the reaction between different species. A time-gate direct fluorescence measurement was conducted for the target material of brass, a Cu–Zn alloy, to validate the simulation results. The simulation results showed good agreement with the experimental results and were able to predict the plume-splitting effect. The ambipolar electric field induced by charged particle distribution difference was studied in detail. It was found that the ambipolar electric field was the dominant cause for the acceleration of ions, which yielded plume splitting during the expansion process. The fully coupled HD model was further used to explore the effect of the beam spot size, laser fluence, and the pulse width on plasma plume splitting.

A high-order simulation method for compressible multiphase flows with condensed-phase explosive detonation in underwater explosions

This study introduces an efficient method designed for the simulation of compressible multiphase flows associated with explosive detonation, primarily in the context of underwater explosions. The proposed approach integrates three core components: the compressible Euler equations, the level-set equation, and the program burn model. Spatial terms of the compressible Euler equations undergo discretization using a fifth-order accuracy weighted essentially non-oscillation reconstruction, while the third-order total variation diminishing Runge–Kutta scheme manages the temporal terms. The level-set method ensures accurate tracking of the multiphase interface. To detail the transition from solid, non-reactive explosives to gaseous detonation products in the condensed charge's detonation reaction zone, the program burn model based on Zeldovich, von Neumann, Doering theory. The efficacy and accuracy of the incorporated program burn model and the multiphase interface capture method are employed through four benchmark tests, exhibiting excellent agreement with previously published data from alternative numerical methods or commercial software. In conclusion, applying the proposed method to four distinct engineering scenarios facilitates a comprehensive understanding of the inherent dynamics associated with detonation and shock wave generation and propagation.

Simulation of temperature profile in fused filament fabrication 3D printing method

PurposeTemperature is a critical factor in the fused filament fabrication (FFF) process, which affects the flow behavior and adhesion of the melted filament and the mechanical properties of the final object. Therefore, modeling and predicting temperature in FFF is crucial for achieving high-quality prints, repeatability, process control and failure prediction. This study aims to investigate the melt deposition and temperature profile in FFF both numerically and experimentally using different Acrylonitrile Butadiene Styrene single-strand specimens. The process parameters, including layer thickness, nozzle temperature and build platform temperature, were varied.Design/methodology/approachCOMSOL Multiphysics software was used to perform numerical simulations of fluid flow and heat transfer for the printed strands. The polymer melt/air interface was tracked using the coupling of continuity equation, equation of motion and the level set equation, and the heat transfer equation was used to simulate the temperature distribution in the deposited strand.FindingsThe numerical results show that increasing the nozzle temperature or layer thickness leads to an increase in temperature at points close to the nozzle, but the bed temperature is the main determinant of the overall layer temperature in low-thickness strands. The experimental temperature profile of the deposited strand was measured using an infrared (IR) thermal imager to validate the numerical results. The comparison between simulation and observed temperature at different points showed that the numerical model accurately predicts heat transfer in the three-dimensional (3D) printing of a single-strand under different conditions. Finally, a parametric analysis was performed to investigate the effect of selected parameters on the thermal history of the printed strand.Originality/valueThe numerical results show that increasing the nozzle temperature or layer thickness leads to an increase in temperature at points close to the nozzle, but the bed temperature is the main determinant of the overall layer temperature in low-thickness strands. The experimental temperature profile of the deposited strand was measured using an IR thermal imager to validate the numerical results. The comparison between simulation and observed temperature at different points showed that the numerical model accurately predicts heat transfer in the 3D printing of a single-strand under different conditions. Finally, a parametric analysis was performed to investigate the effect of selected parameters on the thermal history of the printed strand.

Enhancing the performance of composite phase change materials using novel triply periodic minimal surface structures

Organic phase change materials (PCMs) are widely used in thermal management applications owing to their high latent heats of fusion and low cost. Unfortunately, their poor thermal conductivities severely hamper the thermal transport process. Hence, PCMs are often hybridized with high thermal conductivity materials that promote phase change for time-sensitive applications and potentially increase the power density of thermal energy storage (TES) devices. Optimized thermally conductive networks with high-latent-heat PCM matrices are critical in the thermal management of high heat flux applications. With advancements in additive manufacturing, it has become easier to create complex topologies, such as triply periodic minimal surfaces (TPMS), that can further augment the performance of composite PCMs. TPMS surfaces have shown great promise in improving thermal transport performance but investigations into their thermal properties remain limited. In this study, the thermal performances of PCM infused TPMS structures of wall thicknesses varying from 0.4 mm to 0.8 mm were systematically investigated. The TPMS structures were designed using level-set equations, and each system of structures was fabricated by selective laser melting, a metal additive manufacturing technique. Numerical investigations were performed to understand the phase change dynamics of PCM in TPMS cells and experiments were performed to determine their thermal properties and phase change performance for thermal energy storage applications. Our simulations revealed that for structures with uniform wall thickness of 0.4 mm, the novel Neovius structure resulted in the highest effective thermal conductivity enhancement. For a given thickness, it is thus hypothesized that more complex TPMS topologies have higher surface areas and volumes which enhance their thermal conductivities. However, the complex shape of the Neovius structure could not be further uniformly thickened without self-intersecting geometries. Ultimately, a balance of both the surface area and thickness of the TPMS cell determines the thermal conductivity enhancement. Overall, the 0.6 mm thick IWP cells had the highest thermal conductivity of 13.84 W/m K, which was consistently verified in experiments. This is likely due to the fact that it has a larger surface area than the P structure but is also thicker than the 0.4 mm Neovius structure. To further demonstrate the potential application of our PCM-based TPMS design, low temperature thermal cycling tests were performed. Our results showed the IWP cell structure promoted a more uniform temperature distribution as compared to the P cell. Phase change in the 0.6 mm thick IWP cell structure was completed 1.4 times faster than the 0.8 mm thick P cell and 1.8 times faster than the 0.6 mm thick P cell. The purpose of this study is to provide useful guidelines for the selection of TPMS structures for various applications such as battery packs and spacecraft and to open doors for the co-optimization of materials and device geometry for enhanced energy and power density.

Dynamics of two unequal micron-sized viscous emulsion drops flowing in a two-dimensional constricted microcapillary channel: A numerical study

Emulsions containing unequal-sized droplets are ubiquitous. Compared to the equal-sized emulsion droplet coalescence process in porous media, unequal-sized emulsion droplets show distinctly different dynamics and features when flowing through the capillary channel. In this paper, we numerically studied the flow of two unequal small micron-size emulsion drops flowing in a two-dimensional (2D) half-sine pulse shape constricted tube. The finite element method solves the momentum, continuity, and level set equations. The oil–water interface is tracked using the level set method. Effects of drop size, capillary number (interfacial tension), viscosity ratio, and geometrical shape constriction effects on the droplets' flow behavior across the throat, its flow velocities, and average viscous pressure across the tube were investigated. It is found when there is a significant difference in diameters of the two un-equalized droplets, they will flow separately through the throat, whereas when the two un-equalized droplets’ size is close to each other, they coalescence with each other after passing through the constriction. Higher interfacial tension leads unequal-sized droplets to coalescence more readily while droplets pass without coalescence at low interfacial tension cases. The highest pressure drop is recorded at the critical point where the two unequal-sized droplets begin to merge in the constriction throat. Increasing droplet viscosity reduces the chances of coalescence and increases the average viscous pressure drop hence leading to mobility reduction. In the case of geometrical constrictions, circular-shaped contracted tubes offer the least resistance to fluid flow and are more appropriate for enhanced oil recovery processes. We hope our numerical findings will provide more information to facilitate the enhanced oil recovery process.

Reactor physics characterization of triply periodic minimal surface-based nuclear fuel lattices

Triply periodic minimal surface (TPMS) lattices are receiving substantial attention in numerous engineering fields due to their impressive topology-driven physical characteristics. TPMS lattices are periodic structures of two distinct intertwined volume domains separated by an area-minimizing surface or wall. TPMS lattices have been observed in nature, such as biological membranes, skeletons, block copolymers, sea urchins, butterfly wings, and equipotential surfaces in crystals. Intriguingly, the topology of TPMS lattices can be easily parametrized via level-set equations and thus are heavily numerically and experimentally studied. A significant research effort is currently applying TPMS lattices for heat exchangers and sinks. This paper extends TPMS lattice applications to nuclear reactor fuel designs, with a focus on identifying relevant TPMS geometric parameters controlling neutronics characteristics, such as reactivity, neutron spectrum, and heat removal properties. We found that fuel surface-area-to-volume ratios for TPMS lattices can be two orders of magnitude larger than current cylindrical fuel rods. Further, the selected TPMS lattice and its implicit equation, the unit cell pitch, wall thickness, and structure porosity are design parameters enabling neutronics optimization for both thermal and fast spectrum configurations, paving the way for exceptionally compact and dense nuclear core concepts.

Design Optimization of HTS Field Coils for High Power Density Motors Based on Continuum Sensitivity Analysis

This study proposes an optimization method for designing HTS field coils for high power density motors based on continuum sensitivity analysis. The continuum sensitivity of a superconducting system represents the gradient of the objective function with respect to the superconducting material shape, taking the critical current characteristics into account. The continuum sensitivity was coupled with the level set method through the design velocity field, and the design of the HTS field coil was performed by solving the level set equation. The applicability of the proposed method was verified by designing a 200 kW HTS motor to maximize the torque density. The design optimization provided an optimal solution through the evolution of the topology and shape of the HTS field coil. The optimized geometry was employed to design field coils consisting of racetrack coils wound with HTS tape. Finally, further designs with different cross-sectional areas of the field coils demonstrated that the proposed method can design HTS motors with minimal HTS tape for a given rated power and motor size.

A mass-preserving level set method for simulating 2D/3D fluid flows with deformed interface

This research paper focuses on the development of a novel interface-capturing solver for predicting two-phase flow in 2D/3D Cartesian meshes within the framework of Eulerian approaches. The primary objective is to ensure mass conservation and accurate capture of interface topology. To achieve this, a mass-preserving level set advection equation formulated as a scalar sign-distance function is proposed. The key innovation of the Eulerian solver lies in the incorporation of a scalar speed function for robust reconstruction of the classical level set equation. The effectiveness and reliability of the proposed flow solver in solving incompressible two-phase viscous flow equations are demonstrated through several benchmark problems.

Teaching Effect Evaluation of Music Education Based on Partial Differential Equation Model

Abstract Firstly, this study uses a partial differential equation to conduct experimental research on the teaching effect of music education. Secondly, the convolutional neural network method is used to optimize the weighted factors in the traditional finite volume method. This makes the calculation results of the model more accurate under the condition of coarse particles. It is found that the model can solve Burgers and level set equations accurately and effectively. This method can evaluate the teaching effect of music education more accurately and improve students’ learning products of music.

Response Mitigation of Floating Platform by Porous-Media-Tuned Liquid Dampers

AbstractA porous-media-tuned liquid damper (PMTLD) can serve as an effective and economical dynamic vibration absorber. Placing porous media within a water tank can improve the capacity for energy dissipation and optimize the performance by varying its material properties. Two numerical models are adopted to simulate the sloshing problem in PMTLD and the dynamics of a floating platform in waves. Besides, the effectiveness of response mitigation can be verified numerically. The first potential-based approach employs a mixed-type boundary value problem (BVP) solver and a free-surface particle tracker. This approach not only simulates the inviscid water wave but also includes the nonlinear damping of the PMTLD via a quadratic Forchheimer term. Another equivalent mechanical model is used to reduce the degree-of-freedom of the PMTLD system. The Newmark method is incorporated to solve the rigid-body dynamics. The second viscous approach uses the finite element method (FEM) to spatially discretize the Navier–Stokes (NS) equations and handles the free surface via the volume of fluid (VOF) and the level set (LS) equations. The multiphase simulation is implemented by computational modeling toolkits, Proteus and Chrono, for the fluid and solid phases, respectively. The correlations between potential flow and two-phase NS models are presented. The PMTLD is designed by analogy with the tuned mass damper (TMD). Numerical results show that the PMTLD can effectively reduce the structure's dynamic response in terms of vibration amplitude around resonance. Such damping devices have great potential for offshore platforms and wind turbine design.

Macroscopic and mesoscopic simulation of polystyrene melt in the filling process of injection molding

AbstractIn this article, a fluid flow and heat transfer model which consist of the mass, momentum, and energy conservation equations together with the level set equation and Rolie‐Poly constitution equation is proposed and applied to simulate the rheological behaviors of viscoelastic polystyrene melt in the filling process of injection molding. The macroscopic velocity, pressure, temperature, stress, and mesoscopic molecular conformation of polystyrene melt are shown and analyzed detailedly during the mold filling process. The single‐mode Rolie‐Poly model is employed to describe the stress–strain relationship of polystyrene melt, and the level set method is used to capture the evolution of melt front interface. The governing equations for polymer melt are discretized by the finite volume method on collocated mesh, and solved by the SIMPLE algorithm. Numerical simulations have been implemented based on rectangular cavities with different gate locations, and rectangular and circular inserts. Numerical results indicate that the gate locations, gate numbers, insert size and shape are major factors that affect the rheological behaviors of polystyrene melt in the filling stage of injection molding, and polymer chains experienced higher shear and extension effects in the regions near to mold walls and obstacle. In addition, the length and position of weld lines are discussed in the current study.

An Investigation of the Influence of Viscosity and Printing Parameters on the Extrudate Geometry in the Material Extrusion Process.

The material extrusion process is one of the most popular additive manufacturing processes. The presence of porosity in the MEX printed parts, which ultimately deteriorates the mechanical properties, is one of the main drawbacks of the MEX process. The porosity in the structure is related to the shape of the adjacent beads and overlapping during the material deposition. Due to the deposition nature of the MEX process, the porosity cannot be entirely removed from the printed parts. Understanding the influence of process parameters on material deposition and the rheological properties is crucial to improving the quality of the final product. In this study, the two-phase-flow numerical approach with the level-set equations has been used for the first time to model the material deposition on the moving platform in 3D. The influence of the viscosity and printing parameters, including travel speed, inlet velocity, viscosity, nozzle diameter, and layer height, on the width of the deposited bead has been investigated. The simulation results are validated against experimental measurements with an average error of 5.92%. The width measured by the experimental study shows good agreement with the results of the numerical simulation. The comparison between the results of the 3D numerical simulation and 2D simulation reveals that the 2D simulation is not appropriate and accurate enough to predict the geometry of the deposited bead with the given set of parameter settings. The key novelty of this research paper is the application of the level-set method in a 3D context for material deposition on a moving substrate.

Bayesian Physics Informed Neural Networks for data assimilation and spatio-temporal modelling of wildfires

We apply the Physics Informed Neural Network (PINN) to the problem of wildfire fire-front modelling. We use the PINN to solve the level-set equation, which is a partial differential equation that models a fire-front through the zero-level-set of a level-set function. The result is a PINN that simulates a fire-front as it propagates through the spatio-temporal domain. We show that popular optimisation cost functions used in the literature can result in PINNs that fail to maintain temporal continuity in modelled fire-fronts when there are extreme changes in exogenous forcing variables such as wind direction. We thus propose novel additions to the optimisation cost function that improves temporal continuity under these extreme changes. Furthermore, we develop an approach to perform data assimilation within the PINN such that the PINN predictions are drawn towards observations of the fire-front. Finally, we incorporate our novel approaches into a Bayesian PINN (B-PINN) to provide uncertainty quantification in the fire-front predictions. This is significant as the standard solver, the level-set method, does not naturally offer the capability for data assimilation and uncertainty quantification. Our results show that, with our novel approaches, the B-PINN can produce accurate predictions with high quality uncertainty quantification on real-world data.

Concurrent topology optimization of multiple components sharing partial design domain based on distance regularized parameterized level set method

AbstractIn this article, the distance regularized parameterized level set method (DRPLSM) is proposed, which introduces a specified energy functional into the level set equation as the distance regularization term. Geometric verification shows that the distance regularization scheme keeps the level set function near the structural boundary as an approximate signed distance function. Based on benchmark example of the cantilever beam, it is verified that the DRPLSM can avoid the re‐initialization of level set function in the optimization process and obtain smooth structural boundary. The DRPLSM is applied to multicomponent concurrent topology optimization with partially shared design domain for the first time. It makes the optimization results have smooth structural boundaries. The optimization objective of this algorithm is to minimize the structural compliance of all components. The sharing domain is divided into different design domains according to the strain energy of elements. The optimality of the proposed algorithm for partitioning the sharing domain is verified using the literature example with known optimal solutions. A symmetric structure example, a main and auxiliary optimization structure example and an example with design‐dependent boundary loads show that the proposed concurrent optimization algorithm has wide applicability.

Adjoint-based optimization of two-dimensional Stefan problems

A range of optimization cases of two-dimensional Stefan problems, solved using a tracking-type cost-functional, is presented. A level-set method is used to capture the interface between the liquid and solid phases and an immersed boundary (cut cell) method coupled with an implicit time-advancement scheme is employed to solve the heat equation. A conservative implicit-explicit scheme is then used for solving the level-set transport equation. The resulting numerical framework is validated with respect to existing analytical solutions of the forward Stefan problem. An adjoint-based algorithm is then employed to efficiently compute the gradient used in the optimization algorithm (L-BFGS). The algorithm follows a continuous adjoint framework, where adjoint equations are formally derived using shape calculus and transport theorems. A wide range of control objectives are presented, and the results show that using parameterized boundary actuation leads to effective control strategies in order to suppress interfacial instabilities or to maintain a desired crystal shape.