We propose a nonlinear manifold learning technique based on deep convolutional autoencoders that is appropriate for model order reduction of physical systems in complex geometries. Convolutional neural networks have proven to be highly advantageous for compressing data arising from systems demonstrating a slow-decaying Kolmogorov n-width. However, these networks are restricted to data on structured meshes. Unstructured meshes are often required for performing analyses of real systems with complex geometry. Our custom graph convolution operators based on the available differential operators for a given spatial discretization effectively extend the application space of deep convolutional autoencoders to systems with arbitrarily complex geometry that are typically discretized using unstructured meshes. We propose sets of convolution operators based on the spatial derivative operators for the underlying spatial discretization, making the method particularly well suited to data arising from the solution of partial differential equations. We demonstrate the method using examples from heat transfer and fluid mechanics and show better than an order of magnitude improvement in accuracy over linear methods.
In this short note, we discuss the use of arithmetic averages for the evaluation of viscous coefficients such as temperature and velocity components at a face as required in a cell-centered finite-volume viscous discretization on unstructured grids, and show that second-order accuracy can be achieved even when the arithmetic average is not linearly-exact second-order reconstruction at a face center (e.g., the face center is not located exactly halfway between two adjacent cell centroids) as typical in unstructured grids. Unlike inviscid discretizations, where the solution has to be reconstruct…
This work is an attempt to develop an approximate scheme for estimating the volume-based truncation errors in the finite volume analysis of laminar flows. The volume-based truncation error is the net flow error across the faces of a control volume. Unfortunately, truncation error is not a natural outcome of the finite volume solution and needs to be estimated separately. Previous works in the literature estimate truncation error using either higher order interpolation schemes, higher order discretization schemes, or neglected terms in the discretization scheme. The first two approaches become …
A cell-centered implicit-explicit updated Lagrangian finite volume scheme on unstructured grids is proposed for a unified first order hyperbolic formulation of continuum fluid and solid mechanics. The scheme provably respects the stiff relaxation limits of the continuous model at the fully discrete level, thus it is asymptotic preserving. Furthermore, the GCL is satisfied by a compatible discretization that makes use of a nodal solver to compute vertex-based fluxes that are used both for the motion of the computational mesh as well as for the time evolution of the governing PDEs. Second-order …
Finite difference method was extended to unstructured meshes to solve Euler equations. The spatial discretization is made of two steps. First, numerical fluxes are computed at the middle point of each edge with high order accuracy. In this step, a one-dimensional curvilinear stencil is assembled for each edge to perform one-dimensional fast non-uniform WENO interpolation derived in this paper, which is much easier and faster than multi-dimensional interpolation. The second step is to compute the divergence of fluxes at each vertex from the fluxes at nearby edges and vertices by least square ap…
In this paper we propose the first better than second order accurate method in space and time for the numerical solution of the resistive relativistic magnetohydrodynamics (RRMHD) equations on unstructured meshes in multiple space dimensions. The nonlinear system under consideration is purely hyperbolic and contains a source term, the one for the evolution of the electric field, that becomes stiff for low values of the resistivity. For the spatial discretization we propose to use high order $\PNM$ schemes as introduced in \cite{Dumbser2008} for hyperbolic conservation laws and a high order acc…
Problems of interest in hydrogeology and hydrocarbon resources involve complex heterogeneous geological formations. Such domains are most accurately represented in reservoir simulations by unstructured computational grids. Finite element methods accurately describe flow on unstructured meshes with complex geometries, and their flexible formulation allows implementation on different grid types. In this work, we consider for the first time the challenging problem of fully compositional three-phase flow in 3D unstructured grids, discretized by any combination of tetrahedra, prisms, and hexahedra.…
A conservative primitive variable discrete exterior calculus (DEC) discretization of the Navier-Stokes equations is performed. An existing DEC method (Mohamed, M. S., Hirani, A. N., Samtaney, R. (2016). Discrete exterior calculus discretization of incompressible Navier-Stokes equations over surface simplicial meshes. Journal of Computational Physics, 312, 175-191) is modified to this end, and is extended to include the energy-preserving time integration and the Coriolis force to enhance its applicability to investigate the late time behavior of flows on rotating surfaces, i.e., that of the pla…
Geophysical model domains typically contain irregular, complex fractal-like boundaries and physical processes that act over a wide range of scales. Constructing geographically constrained boundary-conforming spatial discretizations of these domains with flexible use of anisotropically, fully unstructured meshes is a challenge. The problem contains a wide range of scales and a relatively large, heterogeneous constraint parameter space. Approaches are commonly ad hoc, model or application specific and insufficiently described. Development of new spatial domains is frequently time-consuming, hard…
Simulations of complex physical systems are typically realized by discretizing partial differential equations (PDEs) on unstructured meshes. While neural networks have recently been explored for surrogate and reduced order modeling of PDE solutions, they often ignore interactions or hierarchical relations between input features, and process them as concatenated mixtures. We generalize the idea of conditional parameterization -- using trainable functions of input parameters to generate the weights of a neural network, and extend them in a flexible way to encode critical information. Inspired by…
This paper presents a highly robust third-order accurate finite volume weighted essentially non-oscillatory (WENO) method for special relativistic hydrodynamics on unstructured triangular meshes. We rigorously prove that the proposed method is physical-constraint-preserving (PCP), namely, always preserves the positivity of the pressure and the rest-mass density as well as the subluminal constraint on the fluid velocity. The method is built on a highly efficient compact WENO reconstruction on unstructured meshes, a simple PCP limiter, the provably PCP property of the Harten--Lax--van Leer flux,…
The recently proposed high-order TENO scheme [Fu et al., Journal of Computational Physics, 305, pp.333-359] has shown great potential in predicting complex fluids owing to the novel weighting strategy, which ensures the high-order accuracy, the low numerical dissipation, and the sharp shock-capturing capability. However, the applications are still restricted to simple geometries with Cartesian or curvilinear meshes. In this work, a new class of high-order shock-capturing TENO schemes for unstructured meshes are proposed. Similar to the standard TENO schemes and some variants of WENO schemes, t…
As an extension of previous fourth-order compact gas kinetic scheme (GKS) on structured meshes (Ji et al. 2018), this work is about the development of a third-order compact GKS on unstructured meshes for the compressible Euler and Navier-Stokes solutions. Based on the time accurate high-order gas-kinetic evolution solution at a cell interface, the time dependent gas distribution function in GKS provides not only the flux function and its time derivative at a cell interface, but also the time accurate flow variables there at next time level. As a result, besides updating the conservative flow v…
In this paper, a finite volume lattice Boltzmann method (FVLBM) based on cell-center unstructured girds is presented and full studied to simulate the incompressible laminar flows, which is simple modified from the cell-vertex unstructured girds FVLBM proposed by Stiebler et al. [Computers & Fluids, 2006, 35(8): 814-819]. Compared with other complex flux reconstruct methods, the computational cost of present scheme is little and can achieve second-order spatial accuracy, the temporal accuracy is adjustable depending on the temporal discretization methods. Different boundary conditions are illus…
