Time-dependent Heat Equation Research Articles

Why Fourier mode analysis in time is different when studying Schwarz Waveform Relaxation

Schwarz waveform relaxation methods are Schwarz methods applied to evolution problems. Like for steady problems, they are based on an overlapping domain decomposition of the spatial domain, and an iteration which only requires subdomain solutions, now in space-time, to get better and better approximations of the global, monodomain solution. Fourier analysis has been used to study the convergence of both Schwarz and Schwarz waveform relaxation methods. We show here that their convergence is however quite different: for steady problems of diffusive type, Schwarz methods converge linearly, which is also well predicted by Fourier analysis. For a time dependent heat equation however, the Schwarz waveform relaxation algorithm first has a rapid convergence phase, followed by a slow down, and eventually convergence increases again to become superlinear, none of which is predicted by classical Fourier analysis. Introducing a new Fourier analysis combined with kernel estimates, we can explain this behavior for the heat equation. We then generalize our approach to the case of advection reaction diffusion problems. We illustrate all our results with numerical experiments.

A Variational Quantum Linear Solver Application to Discrete Finite-Element Methods.

Finite-element methods are industry standards for finding numerical solutions to partial differential equations. However, the application scale remains pivotal to the practical use of these methods, even for modern-day supercomputers. Large, multi-scale applications, for example, can be limited by their requirement of prohibitively large linear system solutions. It is therefore worthwhile to investigate whether near-term quantum algorithms have the potential for offering any kind of advantage over classical linear solvers. In this study, we investigate the recently proposed variational quantum linear solver (VQLS) for discrete solutions to partial differential equations. This method was found to scale polylogarithmically with the linear system size, and the method can be implemented using shallow quantum circuits on noisy intermediate-scale quantum (NISQ) computers. Herein, we utilize the hybrid VQLS to solve both the steady Poisson equation and the time-dependent heat and wave equations.

Homogenization of the time-dependent heat equation on planar one-dimensional periodic structures

ABSTRACT In this paper we consider the homogenization of a time-dependent heat conduction problem on a planar one-dimensional periodic structure. On the edges of a graph the one-dimensional heat equation is posed, while the Kirchhoff junction condition is applied at all (inner) vertices. Using the two-scale convergence adapted to homogenization of lower-dimensional problems we obtain the limit homogenized problem defined on a two-dimensional domain that is occupied by the mesh when the mesh period δ tends to 0. The homogenized model is given by the classical heat equation with the conductivity tensor depending on the unit cell graph only through the topology of the graph and lengthes of its edges. We show the well-posedness of the limit problem and give a purely algebraic formula for the computation of the homogenized conductivity tensor. The analysis is completed by numerical experiments showing a convergence to the limit problem where the convergence order in δ depends on the unit cell pattern.

An efficient collocation method for long-time simulation of heat and mass transport on evolving surfaces

This paper presents an efficient collocation method which combines the generalized finite difference method (GFDM) with the Krylov deferred correction (KDC) method for the long-time simulation of heat and mass transport on evolving surfaces. The KDC method utilizes a pseudo-spectral-type temporal collocation formulation to discretize the time-dependent surface heat and mass transport equation in each time marching step, where the time derivatives at the collocation points are introduced as the new unknown variables. A low-order time marching scheme is then applied as an effective preconditioner in the Jacobian-Free Newton-Krylov framework to decouple the spatial surface PDEs at different collocation nodes. Each decoupled surface PDE is then solved by the meshless GFDM, where both the continuous-form evolving surfaces defined by parametric equations and discretized-form evolving surfaces composed of point clouds are considered in the GFDM spatial discretization. Numerical experiments show that the combined GFDM-KDC solver is a promising numerical scheme for long-time evolution simulation of heat and mass transport on intractable evolving surfaces.

Fast active thermal cloaking through PDE-constrained optimization and reduced-order modelling

In this paper, we show how to efficiently achieve thermal cloaking from a computational standpoint in several virtual scenarios by controlling a distribution of active heat sources. We frame this problem in the setting of PDE-constrained optimization, where the reference field is the solution of the time-dependent heat equation in the absence of the object to cloak. The optimal control problem then aims at actuating the space–time control field so that the thermal field outside the obstacle is indistinguishable from the reference field. In particular, we consider multiple scenarios where material’s thermal diffusivity, source intensity and obstacle’s temperature are allowed to vary within a user-defined range. To tackle the thermal cloaking problem in a rapid and reliable way, we rely on a parametrized reduced order model built through the reduced basis method, thus entailing huge computational speedups compared with a high-fidelity, full-order model exploiting the finite-element method while dealing both with complex target shapes and disconnected control domains.

Optimal local truncation error method for solution of elasticity problems for heterogeneous materials with irregular interfaces and unfitted Cartesian meshes

The optimal local truncation error method (OLTEM) with unfitted Cartesian meshes was recently developed for PDEs with homogeneous materials on regular and irregular domains as well as for the scalar time-dependent wave and heat equations for heterogeneous materials with irregular interfaces. Here, OLTEM is extended to a system of time-independent elastic PDEs for heterogeneous materials with irregular interfaces and unfitted Cartesian meshes. We show the development of OLTEM for the 2D elasticity equations using compact 9-point stencils that are similar to those for linear quadrilateral finite elements. The interface conditions on the interfaces where the jumps in material properties occur are added to the expression for the local truncation error and do not change the width of the stencils. There are no unknowns on interfaces between different materials; the structure of the global discrete equations is the same for homogeneous and heterogeneous materials. The calculation of the unknown stencil coefficients is based on the minimization of the local truncation error of the stencil equations and yields the optimal second order of accuracy for OLTEM with 9-point stencils on unfitted Cartesian meshes. Numerical experiments for elastic heterogeneous materials with irregular interfaces show that at the same number of degrees of freedom: a) OLTEM with unfitted Cartesian meshes is more accurate than linear finite elements with similar stencils and conformed meshes; b) up to engineering accuracy of OLTEM with unfitted Cartesian meshes is even more computationally efficient than quadratic and cubic finite elements with much wider stencils and conformed meshes. The proposed technique yields accurate numerical results for heterogeneous materials with big contrasts in the material properties of different components. Due to the computational efficiency and trivial unfitted Cartesian meshes that are independent of irregular geometry, the proposed technique does not require remeshing for the shape change of irregular geometry and it will be effective for many design and optimization problems.

Optical detection of defects during laser metal deposition: Simulations and experiment

Laser metal deposition is a rapidly evolving method for additive manufacturing that combines high performance and simplified production routine. Quality of production depends on instrumental design and operational parameters that require constant control during the process. In this work, feasibility of using optical spectroscopy as a control method is studied via modeling and experimentally. A simplified thermal model is developed based on the time-dependent diffusion-conduction heat equation and geometrical light collection into detection optics. Intense light emitted by a laser-heated spot moving across a sample surface is collected and processed to yield the temperature and other temperature-related parameters. In a presence of surface defects the temperature field is distorted in a specific manner that depends on a shape and size of the defect. Optical signals produced by such the distorted temperature fields are simulated and verified experimentally using a 3D metal printer and a sample with artificially carved defects. Three quantities are tested as possible metrics for process monitoring: temperature, integral intensity, and correlation coefficient. The shapes of the simulated signals qualitatively agree with the experimental signals; this allows a cautious inference that optical spectroscopy is capable of detecting a defect and, possibly, predicting its character, e.g. inner or protruding.

Evaluating the solution accuracy of the heat equation by a posteriori error estimation

A produce based on a guaranteed posterior error estimates is proposed to estimate the accuracy of solution of the time-dependent heat equation. The time-dependent heat equation is discretized by the backward Euler scheme in time and conforming finite elements in space. The error between the exact solution and the numeric solution in space is bounded by a guaranteed posterior error estimates which is calculated at every time step. Numerical results demonstrate that the produce can be used to evaluate the accuracy of numerical solutions.

A non-conservation stochastic partial differential equation driven by anisotropic fractional Lévy random field

We study a non-conservation second-order stochastic partial differential equation (SPDE) driven by multi-parameter anisotropic fractional Lévy noise (AFLN) and under different initial and/or boundary conditions. It includes the time-dependent linear heat equation and quasi-linear heat equation under the fractional noise as special cases. Unique existence and expressions of solution to the equation are proved and constructed. An AFLN is defined as the derivative of an anisotropic fractional Lévy random field (AFLRF) in certain sense. Comparing with existing noise systems, our non-Gaussian fractional noises are essentially observed from random disturbances on system accelerations rather than from those on system moving velocities. In the process of proving our claims, there are three folds. First, we consider the AFLRF as the generalized functional of sample paths of a pure jump Lévy process. Second, we build Skorohod integration with respect to the AFLN by white noise approach. Third, by combining this noise analysis method with the conventional PDE solution techniques, we provide solid proofs for our claims.

A new numerical approach to the solution of PDEs with optimal accuracy on irregular domains and Cartesian meshes—part 2: numerical simulations and comparison with FEM

A new numerical approach for the time-dependent wave and heat equations as well as for the time-independent Poisson equation developed in Part 1 is applied to the simulation of 1-D and 2-D test problems on regular and irregular domains. Trivial conforming and non-conforming Cartesian meshes with 3-point stencils in the 1-D case and 9-point stencils in the 2-D case are used in calculations. The numerical solutions of the 1-D wave equation as well as the 2-D wave and heat equations for a simple rectangular plate show that the accuracy of the new approach on non-conforming meshes is practically the same as that on conforming meshes for both the Dirichlet and Neumann boundary conditions. Moreover, very small distances ( $$0.1 h - 10^{-9}h$$ where h is the grid size) between the grid points of a Cartesian mesh and the boundary do not decrease the accuracy of the new technique. The application of the new approach to the 2-D problems on an irregular domain shows that the order of accuracy is close to four for the wave and heat equations and is close to five for the Poisson equation. This is in agreement with the theoretical results of Part 1 of the paper. The comparison of the numerical results obtained by the new approach and by FEM shows that at similar 9-point stencils, the accuracy of the new approach on irregular domains is two orders higher for the wave and heat equations and three orders higher for the Poisson equation than that for the linear finite elements. Moreover, the new approach yields even much more accurate results than the quadratic and cubic finite elements with much wider stencils. An example of a problem with a complex irregular domain that requires a prohibitively large computation time with the finite elements but can be easily solved with the new approach is presented.

A new numerical approach to the solution of PDEs with optimal accuracy on irregular domains and Cartesian meshes—Part 1: the derivations for the wave, heat and Poisson equations in the 1-D and 2-D cases

A new numerical approach for the time-dependent wave and heat equations as well as for the time-independent Poisson equation on irregular domains has been developed. Trivial Cartesian meshes and simple 9-point stencil equations with unknown coefficients are used for 2-D irregular domains. The calculation of the coefficients of the stencil equations is based on the minimization of the local truncation error of the stencil equations and yields the optimal order of accuracy. The treatment of the Dirichlet and Neumann boundary conditions in the new approach is related to the development of high-order boundary conditions with the stencils that include the same or a smaller number of grid points compared to that for the regular 9-point internal stencils. At similar 9-point stencils, the accuracy of the new approach is two orders higher than that for the linear finite elements. The numerical results for irregular domains in Part 2 of the paper also show that at the same number of degrees of freedom, the new approach is even much more accurate than the quadratic and cubic finite elements with much wider stencils. Similar to our recent results on regular domains, the order of the accuracy of the new approach for the Poisson equation on irregular domains with square Cartesian meshes is higher than that with rectangular Cartesian meshes. The new approach can be directly applied to other partial differential equations.

New 25-point stencils with optimal accuracy for 2-D heat transfer problems. Comparison with the quadratic isogeometric elements

A new approach for the increase in the order of accuracy of high order elements used for the time dependent heat equation and for the time independent Poisson equation has been suggested on uniform square and rectangular meshes. It is based on the optimization of the coefficients of the corresponding discrete stencil equation with respect to the local truncation error. By a simple modification of the coefficients of 25-point stencils, the new approach exceeds the accuracy of the quadratic isogeometric elements by four orders for the heat equation and by twelve orders for the Poisson equation. Despite the significant increase in accuracy, the computational costs of the new technique are the same as those for the conventional quadratic isogeometric elements on a given mesh. The numerical examples are in a good agreement with the theoretical results for the new approach and also show that the new approach is much more accurate than the conventional isogeometric elements at the same number of degrees of freedom. Hybrid methods that combine the new stencils with the conventional isogeometric and finite elements and can be applied to irregular domains are also presented.

A Geometric Heat-Flow Theory of Lagrangian Coherent Structures

We consider Lagrangian coherent structures (LCSs) as the boundaries of material subsets whose advective evolution is metastable under weak diffusion. For their detection, we first transform the Eulerian advection–diffusion equation to Lagrangian coordinates, in which it takes the form of a time-dependent diffusion or heat equation. By this coordinate transformation, the reversible effects of advection are separated from the irreversible joint effects of advection and diffusion. In this framework, LCSs express themselves as (boundaries of) metastable sets under the Lagrangian diffusion process. In the case of spatially homogeneous isotropic diffusion, averaging the time-dependent family of Lagrangian diffusion operators yields Froyland’s dynamic Laplacian. In the associated geometric heat equation, the distribution of heat is governed by the dynamically induced intrinsic geometry on the material manifold, to which we refer as the geometry of mixing. We study and visualize this geometry in detail, and discuss connections between geometric features and LCSs viewed as diffusion barriers in two numerical examples. Our approach facilitates the discovery of connections between some prominent methods for coherent structure detection: the dynamic isoperimetry methodology, the variational geometric approaches to elliptic LCSs, a class of graph Laplacian-based methods and the effective diffusivity framework used in physical oceanography.

The treatment of the Neumann boundary conditions for a new numerical approach to the solution of PDEs with optimal accuracy on irregular domains and Cartesian meshes

Recently we have developed a new 3-D numerical approach for the time dependent wave and heat equations as well as for the time independent Poisson equation on irregular domains with the Dirichlet boundary conditions. Its extension to the Neumann boundary conditions that was a big issue due to the presence of normal derivatives along irregular boundaries is considered in this paper. Trivial Cartesian meshes and simple 27-point uniform and nonuniform stencil equations are used with the new approach for 3-D irregular domains. The Neumann boundary conditions are introduced as the known right-hand side into the stencil and do not change the width of the stencil. The calculation of the coefficients of the stencil equations is based on the minimization of the local truncation error of the stencil equations and yields the optimal order of accuracy of the new technique. Very small distances (0.1h−10−9h where h is the grid size) between the grid points of a Cartesian mesh and the boundary do not worsen the accuracy of the new technique. At similar 27-point stencils, the accuracy of the new approach is much higher than that for the linear finite elements. The numerical results for irregular domains show that at the same number of degrees of freedom, the new approach is even much more accurate than the high-order (up to the fourth order) tetrahedral finite elements with much wider stencils. The wave and heat equations can be uniformly treated with the new approach. The order of the time derivative in these equations does not affect the coefficients of the stencil equations of the semi-discrete systems. The new approach can be directly applied to other partial differential equations.

Mode instabilities in Yb:YAG crystalline fiber amplifiers.

Mode instabilities (MI) threshold in the Yb:YAG crystalline fiber amplifier is simulated by a full numerical model. The propagation of signal fields is simulated by the finite-difference beam-propagation method combined with the rate equations, and the time-dependent heat equation is solved by the alternating-direction-implicit method. Considering the strong temperature-dependent laser performance of Yb:YAG, an iterative method is applied to reach the steady state of Yb:YAG, the crystalline fiber amplifier, before the simulation of MI behavior. The simulated MI thresholds in Yb:YAG crystalline fiber amplifiers are found to be at least 28 times of those in Yb-doped silica-glass fiber amplifiers, up to tens of kilowatts. Simulation results show that, in addition to the expected higher thermal conductivity and lower thermo-optic coefficient, strong gain saturation also plays an important role in the high MI threshold of the Yb:YAG crystalline fiber.

Communication on partially heated Aluminum 6063-T83 enclosure: Finite element visualization

To inspect thermal individualities in a complicated surfaces being involved in daily life remains a topic a great interest. In this direction the involved boundary constraints mostly leads to singular outcomes and drags the efforts meaningless. The current article is a fruitful step towards heat transfer visualization in a multiply connected domain material made up of an Aluminum 6063-T83. To be more specific, the surface is partially heated via plane and circular heaters. The whole upper wall and some part of lower wall is retained as cold surface. Both the left and right walls of surface are fitted for an adiabatic condition. The said physical model is translated into mathematical model in terms of time dependent heat equation. The lower wall is fitted with the length of two distinct plane heaters. The hybrid meshing is structured for each case. A tremendous method named with Finite Element Method (FEM) is used to offer the strength of surface temperature for upcoming time intervals. The outcomes via Finite Element Method is shared with the aid of graphical trends. It is trusted that the current article will be helpful for beginners towards learning finite element method and heat transfer aspects subject to solid conductor surfaces.

A new 3-D numerical approach to the solution of PDEs with optimal accuracy on irregular domains and Cartesian meshes

A new 3-D numerical approach for the time dependent wave and heat equations as well as for the time independent Laplace equation on irregular domains with the Dirichlet boundary conditions has been developed. Trivial Cartesian meshes and simple 27-point uniform and nonuniform stencil equations are used for 3-D irregular domains. The calculation of the coefficients of the stencil equations is based on the minimization of the local truncation error of the stencil equations and yields the optimal order of accuracy of the new technique. Very small distances (0.1h−10−9h where h is the grid size) between the grid points of a Cartesian mesh and the boundary do not worsen the accuracy of the new technique. At similar 27-point stencils, the accuracy of the new approach is two orders higher than that for the linear finite elements. The numerical results for irregular domains show that at the same number of degrees of freedom, the new approach is even much more accurate than the high-order (up to the fifth order) tetrahedral finite elements with much wider stencils. The wave and heat equations can be uniformly treated with the new approach. The order of the time derivative in these equations does not affect the coefficients of the stencil equations of the semi-discrete systems. The new approach can be directly applied to other partial differential equations.

Semi-analytical solution of the time-dependent heat equation for three-dimensional anisotropic multi-layered media

In this paper, a semi-analytical solution of the time-dependent heat conduction equation is presented for laterally infinite, multi-layered media with anisotropic thermal conductivities, convective outer boundaries and perfect thermal contact between the layers. Space and time-dependent moving heat sources on the boundary as well as inside the medium are considered. In addition, the outer fluid temperature may also depend on space and time. The derived solutions are successfully compared to numerical solutions. A Python implementation of the solution is made freely available.

Differential Harnack estimates for time-dependent heat equations with potentials in a closed spherical CR 3-manifold

In this paper, we derive differential Harnack estimates for time-dependent heat equations with potentials under the CR Yamabe flow in a closed spherical CR 3-manifold with positive Tanaka–Webster curvature and vanishing torsion. It can be served as a generalization of Li–Yau–Hamilton inequality for the CR Yamabe flow. The differential Harnack estimate will play an important role in the study of the CR Yamabe flow.

English

In specific fields of research such as preservation of historical buildings, medical imaging, geophysics and others, it is of particular interest to perform only a non-intrusive boundary measurements. The idea is to obtain comprehensive information about the material properties inside the considered domain while keeping the test sample intact. This paper is focused on such problems, i.e. synthesizing a physical model of interest with a boundary inverse value technique. The forward model is represented here by time dependent heat equation with transport parameters that are subsequently identified using a modified Calderon problem which is numerically solved by a regularized Gauss-Newton method. The proposed model setup is computationally verified for various domains, loading conditions and material distributions.