Partial Differential Equation Solver Research Articles

Flow induced bifurcation and phase-plane stability analysis of branched nanotubes resting on two parameter foundation in a magnetic environment

This paper explores the nonlinear partial differential equations (PDEs) for transverse and longitudinal vibrations of slightly curved fluid-conveying embedded branched nanotubes operating in a magnetic environment. Using Bernoulli–Euler, Nonlocal and Hamilton theories, equations of motion governing the vibrations of the nanotubes are developed. Thereafter, the developed inimitable equations are solved using the numerical PDE solver and PDE-tool in MATLAB. With the numerical solutions, visualizations and parametric studies were carried out. The results indicate that increasing the branch angle at the downstream decreases quantitative stability. Furthermore, the stability results acquired via simulation point out that the magnetic field has a damping or attenuating influence of about 20%. The solutions as presented in this study match with existing results from previous studies, hence the verifications and validations of this work. It is envisioned that the results obtained will provide better physical insights into the stability criteria for carbon nanotubes and will enhance the designs of nanomechanical carbon nanotubes-based devices that convey fluid and operate in elastic and magnetic environments.

Learning finite element convergence with the Multi-fidelity Graph Neural Network

Machine learning techniques have emerged as potential alternatives to traditional physics-based modeling and partial differential equation solvers. Among these machine learning techniques, Graph Neural Networks (GNNs) simulate physics via graph models; GNNs embed relevant physical features into graph data structures, perform message passing within the graphs, and produce new attributes based on the system’s relationships. Like many machine learning frameworks, GNNs are limited by excessive data generation costs and limited generalizability outside of a narrow training domain. To address these limitations, we introduce the Multi-Fidelity Graph Neural Network (MFGNN), a supervised machine learning framework that uses low-fidelity projections to inform high-fidelity modeling across arbitrary subdomains represented by subgraphs. We implement the MFGNN for two-dimensional elastostatic problems with finite element training data. The MFGNN is trained to produce accurate analysis given low-fidelity evaluations and emulate the convergence behavior of traditional finite element analysis (FEA). Through subdomain abstraction, we also extend the MFGNN as a general model for new boundary conditions and material domains outside of the training domain.

Compute-in-Memory for Numerical Computations.

In recent years, compute-in-memory (CIM) has been extensively studied to improve the energy efficiency of computing by reducing data movement. At present, CIM is frequently used in data-intensive computing. Data-intensive computing applications, such as all kinds of neural networks (NNs) in machine learning (ML), are regarded as ‘soft’ computing tasks. The ‘soft’ computing tasks are computations that can tolerate low computing precision with little accuracy degradation. However, ‘hard’ tasks aimed at numerical computations require high-precision computing and are also accompanied by energy efficiency problems. Numerical computations exist in lots of applications, including partial differential equations (PDEs) and large-scale matrix multiplication. Therefore, it is necessary to study CIM for numerical computations. This article reviews the recent developments of CIM for numerical computations. The different kinds of numerical methods solving partial differential equations and the transformation of matrixes are deduced in detail. This paper also discusses the iterative computation of a large-scale matrix, which tremendously affects the efficiency of numerical computations. The working procedure of the ReRAM-based partial differential equation solver is emphatically introduced. Moreover, other PDEs solvers, and other research about CIM for numerical computations, are also summarized. Finally, prospects and the future of CIM for numerical computations with high accuracy are discussed.

Mathematical modeling of unsteady flow with uniform/non-uniform temperature and magnetic intensity in a half-moon shaped domain

The mathematical modeling of two-dimensional unsteady free convective flow and thermal transport inside a half-moon shaped domain charged in the presence of uniform/non-uniform temperature and magnetic effects with Brownian motion of the nanoparticles has been conducted. Thirty-two types of nanofluids in a combination of various nanoparticles and base fluids having different sizes, shapes, and solid concentrations of nanoparticles are chosen to examine the better performance of heat transfer. The circular boundary is cooled while the diameter boundary is heated with uniform/non-uniform temperature. An external uniform/non-uniform/periodic magnetic field is imposed along diameter. The powerful partial differential equations solver, finite element method of Galerkin type, has been engaged in numerical simulation. The numerical solution's heat transfer mechanism reaches a steady state from the unsteady situation within a very short dimensionless time of about 0.65. The thermal transport rate enhances for increasing buoyancy force whereas decreases with higher magnetic intensity. The uniform thermal condition along the diameter of half-moon gives a higher thermal transport rate compared to non-uniform heating conditions. The non-uniform magnetic field provides greater values of the mean Nusselt number than the uniform field. In addition, the outcomes also predict that a better rate of temperature transport for kerosene-based nanofluid than water-based, ethylene glycol-based, and engine oil-based nanofluid. The heat transfer rate is observed at about 67.86 and 23.78% using Co-Kerosene and Co-water nanofluids, respectively, with an additional 1% nanoparticles volume fraction. The blade shape nanoparticles provide a better heat transfer rate than spherical, cylindrical, brick, and platelet shapes. Small size nanoparticles confirm a higher value of average Nusselt number than big size. Mean Nusselt number increases 22.1 and 5.4% using 1% concentrated Cu-water and Cu-engine oil nanofluid, respectively than base fluid.

Fast Method for Accelerating Convergence of Iterative Partial Differential Equation Solvers by Changing System Matrix to Laplacian Counterpart

In this work, we find that the matrix representing the curl-curl operator in a partial differential equation solver of Maxwell’s equations can be analytically decomposed into a gradient divergence operator and a Laplacian, both of which can be constructed from the mesh information without any need for computation. The curl-curl operator can hence be replaced by the Laplacian to find the divergence-free component of the field solution. The Laplacian is positive definite and well-conditioned. As a result, the convergence of an iterative solution of Maxwell’s equations can be guaranteed, and also significantly accelerated. Based on the finding, we represent the divergence-free component of the unknown field solution by deducting its curl-free component. The curl-free component resides in the nullspace of the curl-curl operator, which is also analytically known from the mesh information no matter it is a regular grid or an unstructured mesh. After the divergence-free component is rapidly solved from a Laplacian counterpart of the original system matrix, the curl-free component can also be solved from a Laplacian matrix, and hence having fast and guaranteed convergence. The total computational cost of the proposed method is simply a small number of sparse matrix-vector multiplications. The proposed method has been successfully applied to solve ill-conditioned onchip, packaging, and antenna radiation problems at both low and high frequencies, involving both inhomogeneous dielectrics and lossy conductors. Numerical experiments have demonstrated its fast and guaranteed convergence, as well as trivial computational cost independent of problem size.

An Energy Conserving Vlasov Solver That Tolerates Coarse Velocity Space Resolutions: Simulation of MMS Reconnection Events

AbstractVlasov solvers that operate on a phase‐space grid are highly accurate but also numerically demanding. Coarse velocity space resolutions, which are largely unproblematic in particle‐in‐cell (PIC) simulations, can lead to numerical heating or oscillations in continuum Vlasov methods. To address this issue, we present a new dual Vlasov solver which is based on an established positivity preserving advection scheme for the update of the distribution function and an energy conserving partial differential equation solver for the kinetic update of mean velocity and temperature. The solvers work together via moment fitting during which the maximum entropy part of the distribution function is replaced by the solution from the partial differential equation solver. This numerical scheme makes continuum Vlasov methods competitive with PIC methods concerning computational cost and enables us to model large scale reconnection in Earth's magnetosphere with a fully kinetic continuum method. The simulation results agree well with measurements by the MMS spacecraft.

A Hermite Method with a Discontinuity Sensor for Hamilton–Jacobi Equations

We present a Hermite interpolation based partial differential equation solver for Hamilton-Jacobi equations. Many Hamilton-Jacobi equations have a nonlinear dependency on the gradient, which gives rise to discontinuities in the derivatives of the solution, resulting in kinks. We built our solver with two goals in mind: 1) high order accuracy in smooth regions and 2) sharp resolution of kinks. To achieve this, we use Hermite interpolation with a smoothness sensor. The degrees-of freedom of Hermite methods are tensor-product Taylor polynomials of degree $m$ in each coordinate direction. The method uses $(m+1)^d$ degrees of freedom per node in $d$-dimensions and achieves an order of accuracy $(2m+1)$ when the solution is smooth. To obtain sharp resolution of kinks, we sense the smoothness of the solution on each cell at each timestep. If the solution is smooth, we march the interpolant forward in time with no modifications. When our method encounters a cell over which the solution is not smooth, it introduces artificial viscosity locally while proceeding normally in smooth regions. We show through numerical experiments that the solver sharply captures kinks once the solution losses continuity in the derivative while achieving $2m+1$ order accuracy in smooth regions.

Optimal Dual Schemes for Adaptive Grid Based Hexmeshing

Hexahedral meshes are a ubiquitous domain for the numerical resolution of partial differential equations. Computing a pure hexahedral mesh from an adaptively refined grid is a prominent approach to automatic hexmeshing, and requires the ability to restore the all hex property around the hanging nodes that arise at the interface between cells having different size. The most advanced tools to accomplish this task are based on mesh dualization. These approaches use topological schemes to regularize the valence of inner vertices and edges, such that dualizing the grid yields a pure hexahedral mesh. In this article, we study in detail the dual approach, and propose four main contributions to it: (i) We enumerate all the possible transitions that dual methods must be able to handle, showing that prior schemes do not natively cover all of them; (ii) We show that schemes are internally asymmetric, therefore not only their construction is ambiguous, but different implementative choices lead to hexahedral meshes with different singular structure; (iii) We explore the combinatorial space of dual schemes, selecting the minimum set that covers all the possible configurations and also yields the simplest singular structure in the output hexmesh; (iv) We enlarge the class of adaptive grids that can be transformed into pure hexahedral meshes, relaxing one of the tight topological requirements imposed by previous approaches. Our extensive experiments show that our transition schemes consistently outperform prior art in terms of ability to converge to a valid solution, amount and distribution of singular mesh edges, and element count. Last but not least, we publicly release our code and reveal a conspicuous amount of technical details that were overlooked in previous literature, lowering an entry barrier that was hard to overcome for practitioners in the field.

Integrating deep neural networks with full-waveform inversion: Reparameterization, regularization, and uncertainty quantification

Full-waveform inversion (FWI) is an accurate imaging approach for modeling the velocity structure by minimizing the misfit between recorded and predicted seismic waveforms. However, the strong nonlinearity of FWI resulting from fitting oscillatory waveforms can trap the optimization in local minima. We have adopted a neural-network-based full-waveform inversion (NNFWI) method that integrates deep neural networks with FWI by representing the velocity model with a generative neural network. Neural networks can naturally introduce spatial correlations as regularization to the generated velocity model, which suppresses noise in the gradients and mitigates local minima. The velocity model generated by neural networks is input to the same partial differential equation (PDE) solvers used in conventional FWI. The gradients of the neural networks and PDEs are calculated using automatic differentiation, which back propagates gradients through the acoustic PDEs and neural network layers to update the weights of the generative neural network. Experiments on 1D velocity models, the Marmousi model, and the 2004 BP model determine that NNFWI can mitigate local minima, especially for imaging high-contrast features such as salt bodies, and it significantly improves the inversion in the presence of noise. Adding dropout layers to the neural network model also allows analyzing the uncertainty of the inversion results through Monte Carlo dropout. NNFWI opens a new pathway to combine deep learning and FWI for exploiting the characteristics of deep neural networks and the high accuracy of PDE solvers. Because NNFWI does not require extra training data and optimization loops, it provides an attractive and straightforward alternative to conventional FWI.

Numerical Simulations of a Fluidized Granular Flow Entry Into Water: Insights Into Modeling Tsunami Generation by Pyroclastic Density Currents

AbstractThe tsunami generation potential of pyroclastic density currents (PDCs) entering the sea is poorly understood, due to limited data and observations. Thus far, tsunami generation by PDCs has been modeled in a similar manner to tsunami generation associated with landslides or debris flows, using two‐layer depth‐averaged approaches. Using the adaptive partial differential equation solver Basilisk and benchmarking with published laboratory experiments, this work explores some of the important parameters not yet accounted for in numerical models of PDC‐generated tsunamis. We use assumptions derived from experimental literature to approximate the granular, basal flow component of a PDC as a dense Newtonian fluid flowing down an inclined plane. This modeling provides insight into how the boundary condition of the slope and the viscosity of the dense granular‐fluid influence the characteristics of the waves generated. It is shown that the boundary condition of the slope has a first‐order impact on the interaction dynamics between the fluidized granular flow and water, as well as the energy transfer from the flow to the generated wave. The experimental physics is captured well in the numerical model, which confirms the underlying assumption of Newtonian fluid‐like behavior in the context of wave generation. The results from this study suggest the importance of considering vertical density and velocity stratification in wave generation models. Furthermore, we demonstrate that granular‐fluids denser than water are capable of shearing the water surface and generating significant amplitude waves, despite vigorous overturning.

Learning the solution operator of parametric partial differential equations with physics-informed DeepONets.

Partial differential equations (PDEs) play a central role in the mathematical analysis and modeling of complex dynamic processes across all corners of science and engineering. Their solution often requires laborious analytical or computational tools, associated with a cost that is markedly amplified when different scenarios need to be investigated, for example, corresponding to different initial or boundary conditions, different inputs, etc. In this work, we introduce physics-informed DeepONets, a deep learning framework for learning the solution operator of arbitrary PDEs, even in the absence of any paired input-output training data. We illustrate the effectiveness of the proposed framework in rapidly predicting the solution of various types of parametric PDEs up to three orders of magnitude faster compared to conventional PDE solvers, setting a previously unexplored paradigm for modeling and simulation of nonlinear and nonequilibrium processes in science and engineering.

Coral: a parallel spectral solver for fluid dynamics and partial differential equations

Coral: a parallel spectral solver for fluid dynamics and partial differential equations

High-performance symbolic-numerics via multiple dispatch

As mathematical computing becomes more democratized in high-level languages, high-performance symbolic-numeric systems are necessary for domain scientists and engineers to get the best performance out of their machine without deep knowledge of code optimization. Naturally, users need different term types either to have different algebraic properties for them, or to use efficient data structures. To this end, we developed Symbolics.jl, an extendable symbolic system which uses dynamic multiple dispatch to change behavior depending on the domain needs. In this work we detail an underlying abstract term interface which allows for speed without sacrificing generality. We show that by formalizing a generic API on actions independent of implementation, we can retroactively add optimized data structures to our system without changing the pre-existing term rewriters. We showcase how this can be used to optimize term construction and give a 113x acceleration on general symbolic transformations. Further, we show that such a generic API allows for complementary term-rewriting implementations. Exploiting this feature, we demonstrate the ability to swap between classical term-rewriting simplifiers and e-graph-based term-rewriting simplifiers. We illustrate how this symbolic system improves numerical computing tasks by showcasing an e-graph ruleset which minimizes the number of CPU cycles during expression evaluation, and demonstrate how it simplifies a real-world reaction-network simulation to halve the runtime. Additionally, we show a reaction-diffusion partial differential equation solver which is able to be automatically converted into symbolic expressions via multiple dispatch tracing, which is subsequently accelerated and parallelized to give a 157x simulation speedup. Together, this presents Symbolics.jl as a next-generation symbolic-numeric computing environment geared towards modeling and simulation.

A Fast and Fully Parallel Analog CMOS Solver for Nonlinear PDEs

A general-purpose analog computing method is proposed to compute the continuous-time solutions of nonlinear partial differential equations (PDEs). The discrete-time difference operator in the standard finite difference time domain (FDTD) method is replaced by continuous-time delay operators that can be realized using analog all-pass filters. The resulting spatially discrete time-continuous (SDTC) update equations are realized using analog circuits which compute continuous-time solutions of the PDE with prescribed initial and boundary conditions. The proposed concept is demonstrated in simulation via an integrated circuit (IC) design of a nonlinear acoustic wave equation solver in 180 nm CMOS technology. Analog arithmetic operations (multiply, scale, and add) are realized in parallel using fully differential op-amps and analog multipliers. The proposed IC computes the PDE solution in parallel at 33 discrete spatial points and has a simulated bandwidth and power consumption of approximately 2 MHz and 3 W, respectively. The performance of the IC is simulated using foundry-supplied device models and quantified using i) the mean squared difference between the circuit simulation results and FDTD simulations, and ii) the noise to signal energy ratio. Acceptable accuracy is obtained, with error metric values varying between -7 and -30 dB for various configurations of the problem. Comparison of the custom analog IC simulations with MATLAB- and C-based FDTD code running on a modern workstation shows an expected average speedup of 205× and 140×, respectively.

Kelvin transformations for simulations on infinite domains

Solving partial differential equations (PDEs) on infinite domains has been a challenging task in physical simulations and geometry processing. We introduce a general technique to transform a PDE problem on an unbounded domain to a PDE problem on a bounded domain. Our method uses the Kelvin Transform, which essentially inverts the distance from the origin. However, naive application of this coordinate mapping can still result in a singularity at the origin in the transformed domain. We show that by factoring the desired solution into the product of an analytically known (asymptotic) component and another function to solve for, the problem can be made continuous and compact, with solutions significantly more efficient and well-conditioned than traditional finite element and Monte Carlo numerical PDE methods on stretched coordinates. Specifically, we show that every Poisson or Laplace equation on an infinite domain is transformed to another Poisson (Laplace) equation on a compact region. In other words, any existing Poisson solver on a bounded domain is readily an infinite domain Poisson solver after being wrapped by our transformation. We demonstrate the integration of our method with finite difference and Monte Carlo PDE solvers, with applications in the fluid pressure solve and simulating electromagnetism, including visualizations of the solar magnetic field. Our transformation technique also applies to the Helmholtz equation whose solutions oscillate out to infinity. After the transformation, the Helmholtz equation becomes a tractable equation on a bounded domain without infinite oscillation. To our knowledge, this is the first time that the Helmholtz equation on an infinite domain is solved on a bounded grid without requiring an artificial absorbing boundary condition.

Kelvin transformations for simulations on infinite domains

Solving partial differential equations (PDEs) on infinite domains has been a challenging task in physical simulations and geometry processing. We introduce a general technique to transform a PDE problem on an unbounded domain to a PDE problem on a bounded domain. Our method uses the Kelvin Transform, which essentially inverts the distance from the origin. However, naive application of this coordinate mapping can still result in a singularity at the origin in the transformed domain. We show that by factoring the desired solution into the product of an analytically known (asymptotic) component and another function to solve for, the problem can be made continuous and compact, with solutions significantly more efficient and well-conditioned than traditional finite element and Monte Carlo numerical PDE methods on stretched coordinates. Specifically, we show that every Poisson or Laplace equation on an infinite domain is transformed to another Poisson (Laplace) equation on a compact region. In other words, any existing Poisson solver on a bounded domain is readily an infinite domain Poisson solver after being wrapped by our transformation. We demonstrate the integration of our method with finite difference and Monte Carlo PDE solvers, with applications in the fluid pressure solve and simulating electromagnetism, including visualizations of the solar magnetic field. Our transformation technique also applies to the Helmholtz equation whose solutions oscillate out to infinity. After the transformation, the Helmholtz equation becomes a tractable equation on a bounded domain without infinite oscillation. To our knowledge, this is the first time that the Helmholtz equation on an infinite domain is solved on a bounded grid without requiring an artificial absorbing boundary condition.

Numerical Analysis on the Storage of Nuclear Waste in Gas-Saturated Deep Coal Seam

Nuclear power has contributed humanity a lot since its successful usage in electricity power generation. According to the global statistics, nuclear power accounts for 16% of the total electricity generation in 2020. However, the rapid development of nuclear power also brings up some problems, in which the storage of nuclear waste is the thorny one. This work carries out a series of modeling and simulation analysis on the geological storage of nuclear waste in a gas-saturated deep coal seam. As the first step, a coupled heat-solid-gas model with three constitutional fields of heat transfer, coal deformation, and gas seepage that based on three governing conservation equations is proposed. The approved mechanical model covers series of interactive influences among temperature change, dual permeability of coal, thermal stress, and gas sorption. As the second step, a finite element numerical model and numerical simulation are developed to analyze the storage of nuclear waste in a gas-saturated deep coal seam based on the partial differential equations (PDE) solver of COMSOL Multiphysics with MATLAB. The numerical simulation is implemented and solved then to draw the following conclusions as the nuclear waste chamber heats up the surrounding coal seam firstly in the initial storage stage of 400 years and then be heated by the far-field reservoir. The initial velocity of gas flow decreases gradually with the increment of distance from the storage chamber. Coal gas flows outward from the central storage chamber to the outer area in the first 100 years when the gas pressure in the region nearby the central storage chamber is higher than that in the far region and flows back then while the temperature in the outer region is higher. The modeling and simulation studies are expected to provide a deep understanding on the geological storage of nuclear waste.

Solving of partial differential equations with distributed order in time using fractional-order Bernoulli-Legendre functions

In this paper, an efficient numerical method is used to provide the approximate solution of distributed-order fractional partial differential equations (DFPDEs). The proposed method is based on the fractional integral operator of fractional-order Bernoulli-Legendre functions and the collocation scheme. In our technique, by approximating functions that appear in the DFPDEs by fractional-order Bernoulli functions in space and fractional-order Legendre functions in time using Gauss-Legendre numerical integration, the under study problem is converted to a system of algebraic equations. This system is solved by using Newton's iterative scheme, and the numerical solution of DFPDEs is obtained. Finally, some numerical experiments are included to show the accuracy, efficiency, and applicability of the proposed method.

A robust solver for elliptic PDEs in 3D complex geometries

We develop a boundary integral equation solver for elliptic partial differential equations on complex 3D geometries. Our method is efficient, high-order accurate and robustly handles complex geometries. A key component is our singular and near-singular layer potential evaluation scheme, hedgehog: a simple extrapolation of the solution along a line to the boundary. We present a series of geometry-processing algorithms required for hedgehog to run efficiently with accuracy guarantees on arbitrary geometries and an adaptive upsampling scheme based on a iteration-free heuristic for quadrature error. We validate the accuracy and performance with a series of numerical tests and compare our approach to a competing local evaluation method.

Deep Nitsche Method: Deep Ritz Method with Essential Boundary Conditions

We propose a new method to deal with the essential boundary conditions encountered in the deep learning-based numerical solvers for partial differential equations. The trial functions representing by deep neural networks are non-interpolatory, which makes the enforcement of the essential boundary conditions a nontrivial matter. Our method resorts to Nitsche's variational formulation to deal with this difficulty, which is consistent, and does not require significant extra computational costs. We prove the error estimate in the energy norm and illustrate the method on several representative problems posed in at most 100 dimension.