Meshless Simulation Research Articles

Physics-informed neural network for simulating magnetic field of permanent magnet

Abstract With the rapid development of deep learning, its application in physical field simulation has been widely concerned, and it has begun to lead a new model of meshless simulation. In this paper, research based on physics-informed neural networks is carried out to solve partial differential equations related to the physical laws of electromagnetism. Then the magnetic field simulation is realized. In this method, the governing equation and the boundary conditions containing physical information are embedded into the neural network loss function as constraints, and the backpropagation of neural networks is realized based on automatic differentiation to solve partial differential equations. The high-precision simulation of tile-shaped and rectangular permanent magnet magnetic fields of permanent magnet motors based on physical information neural network is studied, and the error is within 5%. We consider the simulation of magnetic field in two coordinate systems, and realize the joint training of multiple neural networks in multiple sub-domains and different media.

Phase-field formulated meshless simulation of axisymmetric Rayleigh-Taylor instability problem

A formulation of the immiscible Newtonian two-liquid system with different densities and influenced by gravity is based on the Phase-Field Method (PFM) approach. The solution of the related governing coupled Navier-Stokes (NS) and Cahn-Hillard (CH) equations is structured by the meshless Diffuse Approximate Method (DAM) and Pressure Implicit with Splitting of Operators (PISO). The variable density is involved in all the terms. The related moving boundary problem is handled through single-domain, irregular, fixed node arrangement in Cartesian and axisymmetric coordinates. The meshless DAM uses weighted least squares approximation on overlapping subdomains, polynomial shape functions of second-order and Gaussian weights. This solution procedure has improved stability compared to Chorin's pressure-velocity coupling, previously used in meshless solutions of related problems. The Rayleigh-Taylor instability problem simulations are performed for an Atwood number of 0.76. The DAM parameters (shape parameter of the Gaussian weight function and number of nodes in a local subdomain) are the same as in the authors’ previous studies on single-phase flows. The simulations did not need any upwinding in the range of the simulations. The results compare well with the mesh-based finite volume method studies performed with the open-source code Gerris, Open-source Field Operation and Manipulation (OpenFOAM®) code and previously existing results.

Resolving high-strain-rate scratch behavior of Ti6Al4V in experiment and meshless simulation

The outstanding strength-to-weight ratio and corrosion resistance of titanium have made it the material of choice in the aerospace industry and medicine. The alpha–beta alloy Ti6Al4V is particularly preferred for its excellent mechanical and bio-compatible properties. Despite its advantages, the low thermal conductivity and poor tribological performance of titanium pose significant challenges during manufacturing and in operation. This research offers deep insights into the high strain rate behavior of Ti6Al4V under abrasive load, such as e.g. experienced in machining, by modifying the standard scratch test setup and using optimized Johnson–Cook material parameters to perform Material Point Method (MPM) simulations. The MPM simulations provide accurate predictions of the data gathered through high strain rate scratch experiments. We found an increase in the von Mises stress distribution as well as the normal and tangential forces required to perform a scratch of the same depth as the strain rate increases. The morphology of the scratch profiles also showed an increase in the height of the ridges that form as the scratching speed increases. These findings are in line with the increase in yield strength and work hardening with growing strain rate. This study bridges the gap between simulation models and experimental observations by providing insights for improved machining strategies and surface treatments that can enhance the performance of Ti6Al4V in demanding applications.

Machine learning prediction models for investigating vibration properties of epoxy resin under moisture conditions

Epoxy resins used in engineering applications are commonly exposed to wet environment during intended service life, which causes vibration property degradation and increasing risk of structural failure. In this work, vibration properties of epoxy resin plate under different moisture conditions are predicted with various sizes and boundary conditions using developed machine learning (ML) models. The dataset of epoxy vibration is established first, where values in the dataset are calculated with five moisture contents using previously developed meshless model. The dataset from meshless simulation is used to train ML models of epoxy vibration using six different algorithms, including support vector machine, decision tree, random forest, gradient boosting decision tree, extreme gradient boosting, and artificial neural network. It is found that the prediction model developed using extreme gradient boosting algorithm shows the highest accuracy of 99.9% and strong reliability. Using this model, vibration properties of epoxy resin with a series of sizes and boundary conditions are predicted under various moisture contents from dry case to saturated case, which deepens the understanding of the effects of wet environments on the vibration responses of epoxy resins. The results could be used for analysis of durability of epoxy resin, and the developed ML prediction models contribute to investigating vibration property of epoxy resin under different moisture conditions, which is crucial for ensuring durability of epoxy resin in wet environment.

Elasto-plastic simulation of hot shape rolling of steel by a meshless method

This paper describes the development of meshless simulation of thermomechanics of steel in a continuous hot rolling process using a novel meshless solution procedure. During the process, steel billets at high temperatures are transferred through multiple roll passes, each with a specific groove shape and roll gap. The major outcome of this simulation system is to design the rolling schedule and confirm whether the final shape is inside the acceptable range. For this purpose, a perpendicular 2D slice model approach is used to obtain fast and reasonably accurate simulation results. A return mapping algorithm solves the elasto-plastic material model with linear hardening to obtain displacements, strains and stresses. They are calculated by the local radial basis function collocation method (LRBFCM), which uses the local interpolation of the strong form of partial differential equations. The predefined slice positions are entered into the rolling system, and for each position, the slice contact line and amount of reduction are predicted and used as boundary conditions. Thermal and mechanical models are run sequentially, which is repeated until the final slice position reaches the exit from the last roll. The simulation system was validated with multiple rolling schedules provided by the industrial partner and was successfully used in the steel production plant afterwards. The meshless LRBFCM successfully solves a complex elasto-plastic large deformation problem of hot rolling for the first time.

Phase-field formulated meshless simulation of Rayleigh-Taylor instability problem

The interface between two immiscible Newtonian liquids with different densities and the same viscosity, influenced by gravity, is based on the Phase-Field Method (PFM) formulation. The solution of the related governing coupled Navier-Stokes (NS) and Cahn-Hillard (CH) equations is structured by the meshless Diffuse Approximate Method (DAM) and Pressure Implicit with Splitting of Operators (PISO). The variable density is involved in the inertial and buoyancy terms (non-Boussinesq formulation). The related moving boundary problem is handled through single-domain, irregular, fixed node arrangement in two-dimensional Cartesian coordinates. The meshless DAM uses weighted least squares approximation on overlapping subdomains, polynomial shape functions of second-order and Gaussian weights. Implicit time discretisation is performed for the NS and CH equations in the momentum predictor and Phase-Field (PF) variable corrector steps of PISO, while the momentum corrector steps solve the NS equation explicitly. This solution procedure has improved stability compared to Chorin’s pressure-velocity coupling, previously used in meshless solutions of related problems. The Rayleigh-Taylor instability problem simulations are performed for an Atwood number of 0.76. The DAM parameters (shape parameter of the Gaussian weight function and number of nodes in a local subdomain) are the same as in the author’s previous studies on single-phase flows. The simulations did not need any upwinding in the range of the simulations. The results compare well with the mesh-based finite volume method studies performed with the open-source code Gerris.

Meshless Simulation of Multi-site Radio Frequency Catheter Ablation through the Fragile Points Method.

Computational models for radio frequency catheter ablation (RFCA) of cardiac arrhythmia have been developed and tested in conditions where a single ablation site is considered. However, in reality arrhythmic events are generated at multiple sites which are ablated during treatment. Under such conditions, heat accumulation from several ablations is expected and models should take this effect into account. Moreover, such models are solved using the Finite Element Method which requires a good quality mesh to ensure numerical accuracy. Therefore, clinical application is limited since heat accumulation effects are neglected and numerical accuracy depends on mesh quality. In this work, we propose a novel meshless computational model where tissue heat accumulation from previously ablated sites is taken into account. In this way, we aim to overcome the mesh quality restriction of the Finite Element Method and enable realistic multi-site ablation simulation. We consider a two ablation sites protocol where tissue temperature at the end of the first ablation is used as initial condition for the second ablation. The effect of the time interval between the ablation of the two sites is evaluated. The proposed method demonstrates that previous models that do not account for heat accumulation between ablations may underestimate the tissue heat distribution.Clinical relevance- The proposed computational model may be used to build and update a heat map for ablation guidance taking into account the contribution from previously ablated sites. Being a meshless model, it does not require significant input from the user during preprocessing. Therefore, it is suitable for application in a clinical setting.

Meshless simulation of a macrosegregation benchmark considering the solid motion

We have extended the existing two-dimensional rigid solid phase benchmark for binary substance with the solid phase motion in the present paper. Incompressible laminar Newtonian flow is assumed, and a standard mixture formulation is used for the mass, momentum, energy, and solute transport. A coherency solid motion model accounts for the free-floating grains, assuming that the solid velocity is proportional to the mixture velocity and the liquid fraction. The lever rule is used to describe the mass fractions of the phases. A two-dimensional benchmark is solved using the semi-implicit meshless diffuse approximate method with an adaptive subdomain upwinding strategy. The results of the meshless method are compared to the finite volume method results with a reasonable agreement. The new benchmark results show that the solid motion has an essential effect on the macrosegregation pattern.

Understanding moisture effect on nonlinear vibrations of epoxy thin film via a multiscale simulation

Epoxy thin films have been widely used in microelectromechanical systems, aerospace and civil engineering, which are exposed to external excitations in wet environment that lead to severe nonlinear vibration. In this paper, a multiscale simulation consisting of molecular simulation and meshless simulation is adopted to study moisture effect on nonlinear vibrations of epoxy thin film. In molecular simulations, the cross-linked epoxy molecules with moisture content from 0.0 to 4.0 wt% are constructed. It is measured that mechanical properties of epoxy molecules show an initial enhancement with moisture content from 0.0 to 1.0 wt%, and a subsequent decrease when moisture content increases to 4.0 wt%. With molecular simulation results as inputs, meshless simulations are carried out to investigate epoxy vibration behaviors, where vibration equations of epoxy thin films with investigated moisture contents are constructed and solved. It is determined that fundamental frequencies of epoxy thin films show a similar trend as the variation of mechanical properties. Meanwhile, the level of nonlinear frequency ratio decreases in 1.0 wt% case and subsequently increases up to 4.0 wt%. The vibration behaviors of epoxy thin film as revealed in this work contribute to the prediction of vibration behaviors of epoxy-based applications in wet environment.

Physically Based Shape Matching

AbstractThe shape matching method is a popular approach to simulate deformable objects in interactive applications due to its stability and simplicity. An important feature is that there is no need for a mesh since the method works on arbitrary local groups within a set of particles. A major drawback of shape matching is the fact that it is geometrically motivated and not derived from physical principles which makes calibration difficult. The fact that the method does not conserve volume can yield visual artifacts, e.g. when a tire is compressed but does not bulge.In this paper we present a new meshless simulation method that is related to shape matching but derived from continuous constitutive models. Volume conservation and stiffness can be specified with physical parameters. Further, if the elements of a tetrahedral mesh are used as groups, our method perfectly reproduces FEM based simulations.

Vibration analysis of permanent magnet synchronous motor using coupled finite element analysis and optimized meshless method

Conventional finite element analysis (FEA) performed in electromagnetic-vibration coupling calculation of motor suffers from several significant drawbacks, such as large memory space, high computational cost and heavy reliance on mesh quality for accurate solution. With the traditional meshless method, special attention needs to be paid to correctly impose the boundary condition like FEA. Besides, the matrix is easily prone to be ill-conditioned due to introducing large amount of higher-order basis functions. We propose a novel multi-physical coupling method combining FEA and optimized meshless method. The proposed methodology is further evaluated on the vibration analysis of a 12-slot 10-pole permanent magnet synchronous motor (PMSM). Firstly, 2D stator electromagnetic force is simulated and derived based on the local Jacobian derivative method through FEA. The electromagnetic force spectrum is calculated using FFT analysis and further imported into commercial meshless structural simulation software SimSolid for stator harmonic response analysis. Correct force boundary condition and data mapping between meshless and FEA simulation interface are key to the accuracy of the proposed combined multi-physical modeling methodology. This is achieved by introducing new high-dimensional ramp function in the transition region between FEA and meshless domains, which are defined with the shape functions composed of the FEA and meshless method. This function satisfies the continuity and consistency of the displacement function and ensures the convergence of coupled FEA-meshless method. Subsequently, construction of basis function is key to the establishment of convention meshless discrete equation for the elastic problem of rotating machinery. This is designed by using moving least square theory in cylindrical coordinate system. A harmonic response with meshless method is analyzed by using the mode superposition method to obtain detailed mode shape data, acceleration and displacement distribution of stator. Finally, the tangential continuity and robustness are not well considered in the traditional simulation with FEA coupled meshless method. To mitigate this problem, we propose an optimized meshless method based on modified local basis functions to recalculate the harmonic response motion. Then, the coupling electromagnetic-vibration simulation results of traditional coupled FEA-meshless method, optimized coupled FEA-meshless method and complete FEA coupled method are compared. It is worth noting that the optimized method significantly improves accuracy, robustness and computational speed at the same time. In short, the proposed electromagnetic-structure coupling calculation method provides a novel alternative for the multi-physical coupling calculation of rotating machinery combining FEA and meshless simulation methods.

Meshless simulation and experimental study on forced vibration of rectangular stiffened plate

This paper presents an element-free Galerkin (EFG) method for the forced vibration analysis of stiffened plates based on the first-order shear deformation theory (FSDT), and the obtained results are validated by an experiment. First, a stiffened plate is regarded as a composite structure of a flat plate and some stiffeners, and a series of points are used to discretize the flat plate and stiffeners. Second, the moving least-squares (MLS) approximation is used to obtain the potential energy and kinetic energy of the flat plate and stiffeners, respectively. By imposing the displacement compatible conditions between the plate and the stiffeners, the total potential energy and kinetic energy of the stiffened plate can be obtained. Finally, the equation governing the forced vibration behaviors of the stiffened plate is derived according to the Hamilton's Principle. The Newmark-β method is employed to solve the governing equation. A forced vibration experiment of the stiffened plate is carried out. The effects the number and location of stiffeners, and boundary conditions on the forced vibration behaviors of stiffened plates are investigated as well. Compared with experimental test and the finite element software ABAQUS, the present method shows good convergence characteristics and excellent agreement. The present research also shows that stiffened plates are superior to flat plates and that the resonance frequency of the stiffened plate increases as the number of stiffeners increases. The frequency of the dynamic load has a great influence on the plate structure.

Computer simulation of tumour resection-induced brain deformation by a meshless approach.

Tumour resection requires precise planning and navigation to maximise tumour removal while simultaneously protecting nearby healthy tissues. Neurosurgeons need to know the location of the remaining tumour after partial tumour removal before continuing with the resection. Our approach to the problem uses biomechanical modelling and computer simulation to compute the brain deformations after the tumour is resected. In this study, we use meshless Total Lagrangian explicit dynamics as the solver. The problem geometry is extracted from the patient-specific magnetic resonance imaging (MRI) data and includes the parenchyma, tumour, cerebrospinal fluid and skull. The appropriate non-linear material formulation is used. Loading is performed by imposing intra-operative conditions of gravity and reaction forces between the tumour and surrounding healthy parenchyma tissues. A finite frictionless sliding contact is enforced between the skull (rigid) and parenchyma. The meshless simulation results are compared to intra-operative MRI sections. We also calculate Hausdorff distances between the computed deformed surfaces (ventricles and tumour cavities) and surfaces observed intra-operatively. Over 80% of points on the ventricle surface and 95% of points on the tumour cavity surface were successfully registered (results within the limits of two times the original in-plane resolution of the intra-operative image). Computed results demonstrate the potential for our method in estimating the tissue deformation and tumour boundary during the resection.

Effect of CNT volume fractions on nonlinear vibrations of PMMA/CNT composite plates: A multiscale simulation

The volume fraction of the carbon nanotube (CNT) plays a key role in ensuring the performance of the CNT reinforced polymer composite, especially under the severe vibration, which leads to the resonance and failure of the composite structure. In this paper, a bottom-up multiscale method is adopted to study the effect of the CNT volume fractions on the nonlinear vibration of the poly (methyl methacrylate) (PMMA)/CNT composite. According to the molecular simulation, the longitudinal, transverse and shear moduli of the PMMA/CNT nanocomposites are found to increase with the increasing CNT volume fractions. Substituting the simulated moduli into the extended rule of mixtures (EROM), the efficiency parameters of the PMMA/CNT composite with various CNT volume fractions are derived based on a homogenization approach. The derived efficiency parameters are used in the functionally graded (FG)-based EROM to obtain the expressions of the longitudinal, transverse and shear moduli of the macroscopic composite plate, so as to obtain the constitutive equation for the nonlinear vibrations of the FG-based PMMA/CNT composite plate. The subsequent meshless simulation results demonstrate that the natural frequencies of the FG-based composite plate increase with the increasing volume fractions, whereas the ratios of the nonlinear to linear frequencies decrease. Using the bottom-up multiscale analysis, the macroscopic vibration responses are analyzed for the PMMA/CNT composites with the CNT volume fractions up to 9.0%, which provides a paradigm of the design of the PMMA/CNT composite by considering the CNT volume fraction effect.

Meshless simulation of a lid-driven cavity problem with a non-Newtonian fluid

The purpose of the present paper is to solve the lid-driven cavity problem for a non-Newtonian power-law shear thinning and shear thickening fluid by a meshless method. Results are presented for Re=100 and Re=1000, where different levels of shear thinning and thickening are considered. Furthermore, the lid-driven cavity case is made geometrically more complex by adding several circular-shaped obstacles in the geometry. The Navier-Stokes equations are solved with the local meshless diffuse approximate method. The weighted least squares approximation is structured by using the second-order polynomial basis vector and the Gaussian weight function. The explicit Euler scheme is used to perform the artificial time stepping. The non-incremental pressure correction scheme is used to couple the pressure and the velocity fields. Results are presented in terms of viscosity contours, stream function, velocity vectors, mid-plane velocity and viscosity profiles for the steady state. The numerical method is verified by a comparison with the results obtained from the literature, which are computed with the least squares finite element formulation. Furthermore, an investigation of the node density and node distribution type is performed. A near perfect match with the reference solution and between the structured and unstructured node distribution is found.

Modeling of hypervelocity impact on open cell foam core sandwich panels

Sandwich panels are widely used in the design of unmanned satellites and in addition to having a structural function, can often serve as orbital debris shielding. In this application, sandwich panels with open-cell foam cores have a significant advantage, enabling intensive interaction between the impactor fragments and the foam ligaments which enhances the fragments’ breakdown and reduces their damaging potential. Modeling of this process requires an explicit meso-scale representation of the core material. In this study, realistic geometry models were obtained for 10 ppi and 20 ppi 8% aluminum open-cell foams using X-ray Computed Tomography imaging. They were then converted into meshless simulation models and used in modeling of 6.9 km/s particles impact on foam core sandwich panels. Results of the numerical simulations were compared with experimental data reported by NASA in terms of the panels’ core and rear facesheet damage.

Simulation of hardened cement degradation and estimation of uncertainty in predicted failure times with peridynamics

Modeling the degradation of cement-based infrastructure due to aqueous environmental conditions continues to be a challenge. In order to develop a capability to predict concrete infrastructure failure due to chemical degradation, we created a chemomechanical model of the effects of long-term water exposure on cement paste. The model couples the mechanical static equilibrium balance with reactive–diffusive transport and incorporates fracture and failure via peridynamics (a meshless simulation method). The model includes fundamental aspects of degradation of ordinary Portland cement (OPC) paste, including the observed softening, reduced toughness, and shrinkage of the cement paste, and increased reactivity and transport with water induced degradation. This version of the model focuses on the first stage of cement paste decalcification, the dissolution of portlandite. Given unknowns in the cement paste degradation process and the cost of uncertainty quantification (UQ), we adopt a minimally complex model in two dimensions (2D) in order to perform sensitivity analysis and UQ. We calibrate the model to existing experimental data using simulations of common tests such as flexure, compression and diffusion. Then we calculate the global sensitivity and uncertainty of predicted failure times based on variation of eleven unique and fundamental material properties. We observed particularly strong sensitivities to the diffusion coefficient, the reaction rate, and the shrinkage with degradation. Also, the predicted time of first fracture is highly correlated with the time to total failure in compression, which implies fracture can indicate impending degradation induced failure; however, the distributions of the two events overlap so the lead time may be minimal. Extension of the model to include the multiple reactions that describe complete degradation, viscous relaxation, post-peak load mechanisms, and to three dimensions to explore the interactions of complex fracture patterns evoked by more realistic geometry is straightforward and ongoing.

A Moving Least Squares Based Meshless Element-Free Galerkin Method for the Coupled Simulation of Groundwater Flow and Contaminant Transport in an Aquifer

In this study, a meshless simulation model based on the element-free Galerkin method (EFGM) is proposed for the coupled simulation of groundwater flow and contaminant transport (GFCT) in the two-dimensions. EFGM discretizes the governing GFCT equations using the Galerkin integral approach and shape functions generated through the moving least squares (MLS) approximation method. It results into the stable system of equations for estimating the groundwater head and contaminant concentration values. EFGM computes the several global matrices using the spatial information of the scattered and unconnected field nodes and hence, terminates the meshing procedure unlike the finite difference (FDM) and finite element (FEM) methods. The Kronecker delta function property is missing in EFGM shape functions and hence, the Dirichlet boundary conditions are enforced with some special methods. We have used the penalty method to include them into the presented EFGM model. This method is straightforward and computationally useful for the real field simulations. The proposed model is first validated using both the one- and two-dimensional analytical solutions. The developed coupled model is then applied to solve the hypothetical and real field problems. EFGM results for both the problems are compared with the MODFLOW and MT3DMS results for the head and concentration values respectively. EFGM results are found to be satisfactory for both the cases, and it shows the potential of the proposed approach for the coupled GFCT simulations.

Optimal design of in-situ bioremediation system using the meshless element-free Galerkin method and particle swarm optimization

An optimal design of in-situ bioremediation system for contaminated aquifer sites mainly depends on the injection/extraction well locations and their pumping rates. Previous studies have used discretization approaches such as finite difference method (FDM) or finite element method (FEM) to solve the equations of groundwater flow and contaminant transport (GFCT) that characterize the bioremediation processes. In these numerical models, well locations can only lie at the grid nodes and therefore, the locations other than the grid nodes remain unexplored for the optimal in-situ bioremediation. To explore such locations, in this study a meshless simulation model called BIOEFGM is developed using the element-free Galerkin method (EFGM) and coupled with the particle swarm optimization (PSO). BIOEFGM model provides flexibility in adding injection/extraction wells anywhere in the computational domain during the entire optimization procedure. However, FDM/FEM requires meshing and re-meshing of the computational domain for each set of wells and it increases the computational efforts significantly for solving the optimization problem. In this paper, the proposed BIOEFGM-PSO simulation-optimization (S/O) model is first applied to a well-known bioremediation problem and then to a field type large aquifer problem. The simulation results of BIOEFGM are verified with those from the RT3D simulations. The estimated bioremediation cost from the BIOEFGM-PSO model for the first problem is found to be lesser in comparison to the same calculated with different S/O models in the previous studies. The results of this problem also show that optimized in-situ bioremediation system designed by proposed S/O model takes less remediation time and also favor effective biodegradation of contaminants. For both the problems, BIOEFGM-PSO identified optimal well locations at positions other than that of discretized nodes and it leads to efficient bioremediation. It indicates the effectiveness of the BIOEFGM-PSO model and therefore, can be applied for designing the better optimized in-situ bioremediation systems for field problems.

Multi-Physics and Multi-Scale Meshless Simulation System for Direct-Chill Casting of Aluminium Alloys

This paper represents an overview of the elements of the user-friendly simulation system, developed for computational analysis and optimization of the quality and productivity of the electromagnetically direct-chill cast semi-products from aluminium alloys. The system also allows the computational estimation of the design changes of the casting equipment. To achieve this goal, the electromagnetic and the thermofluid process parameters are coupled to the evolution of Lorentz force, temperature, velocity, concentration, strain and stress fields as well as microstructure evolution. This forms a multi-physics and multi-scale problem of great complexity, which has not been demonstrated before. The macroscopic fluid mechanics, solid mechanics, and electromagnetic solution framework is based on local strong-form meshless formulation, involving the radial basis functions and monomials as trial functions, and local collocation or weighted least squares approximation. It is coupled to the micro-scale by incorporating the point automata solution concept. The entire macro-micro solution concept does not require meshing and space integration. The solution procedure can be easily and efficiently automatically adapted in node redistribution and/or refinement sense, which is of utmost importance when coping with fields exhibiting sharp gradients, which occur in the phase-change problems. The simulation system is coded from scratch in modern Fortran. The elements of the experimental validation of the system and the demonstration of its use for round billet casting in IMPOL Aluminium Industry are shown.