Solution Of Time-dependent Problems Research Articles

The neural particle method – An updated Lagrangian physics informed neural network for computational fluid dynamics

Today numerical simulation is indispensable in industrial design processes. It can replace cost and time intensive experiments and even reduce the need for prototypes. While products designed with the aid of numerical simulation undergo continuous improvement, this must also be true for numerical simulation techniques themselves. Up to date, no general purpose numerical method is available which can accurately resolve a variety of physics ranging from fluid to solid mechanics including large deformations and free surface flow phenomena. These complex multi-physics problems occur for example in Additive Manufacturing processes. In this sense, the recent developments in Machine Learning display promise for numerical simulation. It has recently been shown that instead of solving a system of equations as in standard numerical methods, a neural network can be trained solely based on initial and boundary conditions. Neural networks are smooth, differentiable functions that can be used as a global ansatz for Partial Differential Equations (PDEs). While this idea dates back to more than 20 years (Lagaris et al., 1998), it is only recently that an approach for the solution of time dependent problems has been developed (Raissi et al., 2019). With the latter, implicit Runge–Kutta schemes with unprecedented high order have been constructed to solve scalar-valued PDEs. We build on the aforementioned work in order to develop an Updated Lagrangian method for the solution of incompressible free surface flow subject to the inviscid Euler equations. The method is straightforward to implement and does not require any specific algorithmic treatment which is usually necessary to accurately resolve the incompressibility constraint. Due to its meshfree character, we will name it the Neural Particle Method (NPM). It will be demonstrated that the NPM remains stable and accurate even if the location of discretization points is highly irregular.

Physics Informed Extreme Learning Machine (PIELM)–A rapid method for the numerical solution of partial differential equations

There has been rapid progress recently on the application of deep networks to the solution of partial differential equations, collectively labeled as Physics Informed Neural Networks (PINNs). In this paper, We develop Physics Informed Extreme Learning Machine (PIELM), a rapid version of PINNs which can be applied to stationary and time-dependent linear partial differential equations. We demonstrate that PIELM matches or exceeds the accuracy of PINNs on a range of problems. We also discuss the limitations of neural network-based approaches, including our PIELM, in the solution of PDEs on large domains and suggest an extension, a distributed version of our algorithm -- DPIELM. We show that DPIELM produces excellent results comparable to conventional numerical techniques in the solution of time-dependent problems. Collectively, this work contributes towards making the use of neural networks in the solution of partial differential equations in complex domains as a competitive alternative to conventional discretization techniques.

Numerical Solution of Time-Dependent Problems with Different Time Scales

Problems for time-dependent equations in which processes are characterized by different time scales are studied. Parts of the equations describing fast and slow processes are distinguished. The basic features of such problems related to their approximation are taken into account using finer time grids for fast processes. The construction and analysis of inhomogeneous time approximations is based on the theory of additive operator–difference schemes (splitting schemes). To solve time-dependent problems with different time scales, componentwise splitting schemes and vector additive schemes are used. The capabilities of the proposed schemes are illustrated by numerical examples for the time-dependent convection–diffusion problem. If convection is dominant, the convective transfer is computed on a finer time grid.

Adaptive compression of large vectors

Numerical algorithms for elliptic partial differential equations frequently employ error estimators and adaptive mesh refinement strategies in order to reduce the computational cost. We can extend these techniques to general vectors by splitting the vectors into a hierarchically organized partition of subsets and using appropriate bases to represent the corresponding parts of the vectors. This leads to the concept of \emph{hierarchical vectors}. A hierarchical vector with $m$ subsets and bases of rank $k$ requires $mk$ units of storage, and typical operations like the evaluation of norms and inner products or linear updates can be carried out in $\mathcal{O}(mk^2)$ operations. Using an auxiliary basis, the product of a hierarchical vector and an $\mathcal{H}^2$-matrix can also be computed in $\mathcal{O}(mk^2)$ operations, and if the result admits an approximation with $\widetilde m$ subsets in the original basis, this approximation can be obtained in $\mathcal{O}((m+\widetilde m)k^2)$ operations. Since it is possible to compute the corresponding approximation error exactly, sophisticated error control strategies can be used to ensure the optimal compression. Possible applications of hierarchical vectors include the approximation of eigenvectors and the solution of time-dependent problems with moving local irregularities.

Anisotropic hp-adaptive discontinuous Galerkin method for the numerical solution of time dependent PDEs

We deal with the numerical solution of time dependent problems with the aid of anisotropic hp-grids. We present an algorithm which generates a sequence of anisotropic triangular grids and the corresponding polynomial approximation degrees in such a way that the interpolation error measured in the discrete L∞(0,T;Lq(Ω))-norm (q∈[1,∞] and Ω⊂R2) is under a given tolerance and the number of degrees of freedom is as small as possible. The efficiency of the algorithm is demonstrated by numerical experiments.

The Effect of Time-Stepping on the Accuracy of Green Element Formulation of Unsteady Convective Transport

Despite the significant number of boundary element method (BEM) solutions of time-dependent problems, certain concerns still need to be addressed. Foremost among these is the impact of different time discretization schemes on the accuracy of BEM modeling. Although very accurate for steady-state problems, the boundary element methods more often than not are computationally challenged when applied to transient problems. For the work reported herein, we investigate the level of accuracy achieved with different time-discretization schemes for the Green element method (GEM) solution of the unsteady convective transport equation. The Green element method (a modified BEM formulation) solves the boundary integral theory (A Fredholm integral equation of the second kind) on a generic element of the problem domain in a way that is typical of the finite element method (FEM). In this integration process a new system of discrete equations is produced which is banded and hence amenable to matrix manipulations. This is subsequently deployed to investigate the proper resolution in both space and time for the chosen transient 1D transport problems especially those involving shock wave propagation and different types of boundary conditions. It is found that for three out of the four numerical models developed in this study, the new system of discrete element equations generated for both space and temporal domains exhibits accurate characteristics even for cases involving advection-dominant transport. And for all the cases considered, the overall performance relies heavily on the temporal discretization scheme adopted.

Equations of the surface harmonics method for solving time-dependent neutron transport problems and their verification

Time-dependent equations of the surface harmonics method (SHM) are obtained for planar one-dimensional geometry. The equations are verified by calculations of test problems from Benchmark Problem Book ANL-7416, and the capabilities and efficiency of applying the SHM for solving the time-dependent neutron transport equation in the diffusion approximation are demonstrated. The results of the work show that the implementation of the SHG for full-scale computations will make possible substantial progress in the efficient solution of time-dependent problems of neutron transport in nuclear reactors.

Local moving least square - one-dimensional IRBFN technique: part 11 - unsteady incompressible viscous flows

In this study, local moving least square - one dimensional integrated radial basis function network (LMLS-1D-IRBFN) method is presented and demonstrated with the solution of time-dependent problems such as Burgers' equation, unsteady flow past a square cylinder in a horizontal channel and unsteady flow past a circular cylinder. The present method makes use of the partition of unity concept to combine the moving least square (MLS) and one-dimensional integrated radial basis function network (1D-IRBFN) techniques in a new approach. This approach offers the same order of accuracy as its global counterpart, the 1D-IRBFN method, while the system matrix is more sparse than that of the 1D-IRBFN, which helps reduce the computational cost significantly. For fluid flow problems, the diffusion terms are discretised by using LMLS-1D-IRBFN method, while the convection terms are explicitly calculated by using 1D-IRBFN method. The present numerical procedure is combined with a domain decomposition technique to handle largescale problems. The numerical results obtained are in good agreement with other published results in the literature.

Experimental and numerical investigation of separated flows in subsonic diffusers

The results of a special investigation of the diffuser flowfield are presented for two models of curvilinear diffuser channels with annular and rectangular cross-sections. The flow is visualized and the total pressure fields are measured by means of low-inertia transducers. At the same time, the flows are numerically calculated using commercial programs, together with codes developed by the authors. In these calculations the stationary and time-dependent Reynolds equations closed by different turbulence models, as well as the time-dependent Navier-Stokes equations, were integrated. A considerable difference between the measured data and the results of the numerical calculations in the stationary formulation is found to exist. At the same time, it has been possible to describe the occurrence of spatial inhomogeneities, the flow pattern, and the level of the experimentally observed aerodynamic losses on the basis of the solution of time-dependent problems.

On the solution of time-dependent problems

The time-dependent radiative transfer problems involving non-equilibrium coupling to the material temperature to differential equation and ballistic-diffusive equation have been solved by means of two different techniques, namely, flux-limited approach and maximum entropy method. The behaviour of the radiative intensity is shown graphically. Knowing the radiative intensity allows us to calculate directly some physical parameters such as the reflection function and heat flux that are numerically computed.

Solving time-dependent engineering problems with multiquadrics

This paper deals with the solution of time-dependent problems. The multiquadric radial basis function method is formulated, with a new approach for transient problems. One- and two-dimensional problems are considered. The forward difference and the Crank–Nicolson time-marching schemes for parabolic cases are considered. The central difference integration method of the Newmark family is considered for hyperbolic problems. The method proves its accuracy in four numerical examples.

A Collocation Solution For Burgers Equation Using Quadratic B-Spline Finite Elements

This paper discusses solving one of the important equations in Fluid Dynamics; which is the one-dimensional Burgers equation. For that purpose, we use quadratic B-spline finite elements within a Collocation method. It is shown that this method is capable of solving Burgers equation accurately for values of viscosity from small to large. Numerical results are obtained for test problems to show the behavior of the solutions of time dependent problems. Computed results are compared with other numerical results.

Practical Aspects of Formulation and Solution of Moving Mesh Partial Differential Equations

Moving mesh partial differential equations (MMPDEs) are used in the MMPDE moving mesh method to generate adaptive moving meshes for the numerical solution of time dependent problems. How MMPDEs are formulated and solved is crucial to the efficiency and robustness of the method. In this paper, several practical aspects of formulating and solving MMPDEs are studied. They include spatial balance, scaling invariance, effective control of mesh concentration, bounds on time steps, multiple sub-steps, and two-level mesh movement. Numerical results are also given.

Spectral Laguerre method for the approximate solution of time dependent problems

This paper presents some efficient algorithms based on the spectral Laguerre approximations of temporal derivatives for time dependent problems.

Accurate continuous numerical solutions of time dependent mixed partial differential problems

In this paper, a separation of variables method for the solution of time dependent problems of the type u tt = c(t)u xx , 0 < x < p, t > 0, subject to u(0, t) = u(p, t) = 0 and u( x, 0) = ƒ( x}, u t ( x, 0) = g(x) is developed. First, an exact series solution of the problem is given. Given an admissible error ε > 0, and a bounded domain D(T) = {( x, t); 0 ≤ x ≤ p, 0 ≤ t ≤ T, a continuous numerical solution is constructed so that the approximation error is uniformly upper bounded by ε in D(T)

On corner singularities of solutions to mixed boundary-value problems for second-order elliptic and parabolic equations

In this paper we prove a decomposition formula for solutions of mixed boundary-value problems for elliptic and parabolic equations into regular and singular parts, which provides an explicit description of the behaviour of the solution near the corners of the boundary and also, in parabolic cases, shows how the singularity due to the corners varies in time. We also give some sufficient conditions for the regularity of the solution of time-dependent problems.

Comparison of One- and Three-Dimensional Models for Tracer Evolution in Tissue Cylinders

Eigenvalues of the time-reduced operator for a one-dimensional model for diffusion and transport of tracers in tissue cylinders are shown to be limits of some of the eigenvalues of the corresponding three-dimensional operator under an appropriate limiting process. The proof depends on the constructionof approximate eigenfunctions for the three-dimensional operators from eigenfunctions of the one-dimensional operator. Solutions of time-dependent problems with inhomogeneous boundary data are shownto converge to solutions of a one-dimensional time-dependent problem.

Finite element solution of time-dependent problems

In this paper the authors compare the finite element solution of some illustrative time-dependent problems involving parabolic and hyperbolic partial differential equations with alternative solution methods.

Solution of parabolic and hyperbolic time dependent problems using boundary elements

The present paper reviews some recent advances in the solution of time dependent problems by using boundary elements.

The Solution of Time Dependent Problems by Galerkin Methods

The Solution of Time Dependent Problems by Galerkin Methods