Connections between peridynamics and graph Laplacian for diffusion problems

Formulations for diffusion processes based on graph Laplacian kernels have been recently used to solve linear transient heat transfer problems with insulated boundary conditions by way of a spectral-based semi-analytical approach. This has been called the “spectral graph” (SG) approach. In this paper we show that the meshfree discretizations for corresponding peridynamic models have a similar graph structure and lead to the same general equation as the SG approach. In this sense, the SG approach can be seen as a particular case of a peridynamic formulation. For the transient heat diffusion, we explain some differences between the SG approach and peridynamics, related to calibration and discretization procedures. We use a 1D heat diffusion example to highlight some limitations the spectral-based semi-analytical method in the SG approach has compared with the direct time-integration normally used in computing solutions to peridynamic models. We also introduce an extension of the semi-analytical approach to diffusion problems with Dirichlet boundary conditions.

Convergence and error analysis of an automatically differentiated finite volume based heat conduction code

Purpose This paper aims to investigate the convergence and error properties of a finite volume-based heat conduction code that uses automatic differentiation to evaluate derivatives of solutions outputs with respect to arbitrary solution input(s). A problem involving conduction in a plane wall with convection at its surfaces is used as a test problem, as it has an analytical solution, and the error can be evaluated directly. Design/methodology/approach The finite volume method is used to discretize the transient heat diffusion equation with constant thermophysical properties. The discretized problem is then linearized, which results in two linear systems; one for the primary solution field and one for the secondary field, representing the derivative of the primary field with respect to the selected input(s). Derivatives required in the formation of the secondary linear system are obtained by automatic differentiation using an operator overloading and templating approach in C++. Findings The temporal and spatial discretization error for the derivative solution follows the same order of accuracy as the primary solution. Second-order accuracy of the spatial and temporal discretization schemes is confirmed for both primary and secondary problems using both orthogonal and non-orthogonal grids. However, it has been found that for non-orthogonal cases, there is a limit to the error reduction, which is concluded to be a result of errors in the Gauss-based gradient reconstruction method. Originality/value The convergence and error properties of derivative solutions obtained by forward mode automatic differentiation of finite volume-based codes have not been previously investigated.

Steady-state heat conduction in a medium with spatial non-singular fading memory: Derivation of Caputo-Fabrizio space-fractional derivative from Cattaneo concept with Jeffrey`s Kernel and analytical solutions

Starting from the Cattaneo constitutive relation with a Jeffrey's kernel the derivation of a transient heat diffusion equation with relaxation term expressed through the Caputo-Fabrizio time fractional derivative has been developed. This approach allows seeing the physical back ground of the newly defined Caputo-Fabrizio time fractional derivative and demonstrates how other constitutive equations could be modified with non-singular fading memories.