Discovering interpretable physical models using symbolic regression and discrete exterior calculus

The physical models and numerical methodologies of Computational Fluid Dynamics (CFD) are historically linked to the concept of continuous medium and to analysis where continuity, derivation and integration are defined as limits at a point. The first consequence is the need to extend these notions in a multidimensional space by establishing a global inertial frame of reference in order to project the variables there. In recent decades, the emergence of methodologies based on differential geometry or exterior calculus has changed the point of view by starting with the creation of entangled polygonal and polyhedral structures where the variables are located. Mimetic methods and Discrete Exterior Calculus, notably, have intrinsic conservation properties which make them very efficient for solving fluid dynamics equations. The natural extension of this discrete vision relates to the derivation of the equations of mechanics by abandoning the notion of continuous medium. The Galilean frame of reference is replaced by a local frame of reference composed of an oriented segment where the acceleration of the material medium or of a particle is defined. The extension to a higher dimensional space is carried from cause to effect, from one local structure to another. The conservation of acceleration over a segment and the Helmholtz–Hodge decomposition are two essential principles adopted for the derivation of a discrete law of motion. As the fields covered by CFD are increasingly broad, it is natural to return to the deeper meaning of physical phenomena to try a new research or new path which would preserve the properties of current formulations.

I: Elementary Properties of Deformations and Stresses. Deformations and Strain. General Properties of Stresses. Stressed State at a Point. Isotropic Elastic Medium, Equation of State, and the Murnaghan Formulas. Viscous Stresses in an Isotropic Medium. Strain Tensors and Distortion Tensor. Murnaghan Formulas in the Tensor Form and Small Deformations. Moving Medium. Murnaghan Formulas. II: Effective Elastic Deformation. Relaxation of Stresses. Equations for the Metric Effective Elastic Deformation Tensor. Compatibility Conditions. Description of Relaxation of Shear Stresses by Equations for Effective Distortion. Defects of Crystal Lattice. Review of Notions. Dislocations. III: Differential Equations of Dynamical Processes. Conservation Laws. Consequences of Conservation Laws. Deformation of a Layer and a Bar. Equations for One-Dimensional Nonstationary Processes and Structure of Stationary Waves in a Maxwell Medium. IV: Well Posedness of Differential Equations and Thermodynamics. Well Posedness Conditions for Equations of Elasticity. Equations of Linear Elasticity as Symmetric Hyperbolic Equations. Symmetrization of Equations of Gas Dynamics. Influence of Small Viscosity on the Behavior of Solutions. Well Posedness Stability and Postulates of Phenomenological Thermodynamics. V: Multi-Dimensional Thermodynamically Compatible Conservation Laws. Gas Dynamics Equations and Magnetic Hydrodynamics Equations. Equations of Elasticity. Structure of Multi Dimensional Equations. Symmetric Hyperbolicity in the Three-Dimensional Case. Appendix: Structure of Thermodynamically Compatible Systems S.K. Godunov. Mathematical Aspects. The Simplest Galilei-Invariant Thermodynamically Compatible Systems. Methods of Constructing Equations. Some Facts of the Theory of Representations of Orthogonal Transformations. The Clebsch-Gordan Coefficients. Orthogonal Invariants. Literature. Subject Index.

This chapter provides a tutorial for the multigrid solver module in Diffpack. With the MGtools class or the NonLinMGtools class, a standard Diffpack finite element application code can be equipped with a flexible multigrid solver by adding about 10 lines of extra code. The applications covered here are the Poisson equation, more general elliptic problems with anisotropic or jumping coefficients and various boundary conditions, the equation of linear elasticity, and a nonlinear Poisson-like problem. KeywordsCoarse GridFine GridMultigrid MethodGrid LevelMenu ItemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Accurate simulation of atmospheric flow in weather and climate prediction models requires the discretization of the governing equations to have a number of desirable properties. Although these properties can be achieved relatively straightforwardly on a latitude-longitude grid, they are much more challenging on the various quasi-uniform spherical grids that are now under consideration. A recently developed scheme—called TRiSK—has these desirable properties on grids that have an orthogonal dual. The present work extends the TRiSK scheme into a more general framework suitable for grids that have a nonorthogonal dual, such as the equiangular cubed sphere. We also show that this framework fits within the wider framework of mimetic discretizations and discrete exterior calculus. One key ingredient is the definition of certain mapping operators that are discrete analogues of the Hodge star operator, enabling the definition of a compatible inner product. Discrete Coriolis terms are also included within the mimet...

We revisit the theory of Discrete Exterior Calculus (DEC) in 2D for general triangulations, relying only on Vector Calculus and Matrix Algebra. We present DEC numerical solutions of the Poisson equation and compare them against those found using the Finite Element Method with linear elements (FEML).

Chapter 1 - Elastic crack problems, fracture mechanics, equations of elasticity and finite-part integrals

A multitude of industries, including energy and process engineering, as well as academia are researching and utilizing new fluid substances to further the aim of sustainability. Knowledge of the thermodynamic properties of these substances is a prerequisite, if they are to be utilized to their fullest potential. To date, the way to acquire reliable knowledge of the thermodynamic behavior is through measurements. The ensuing experimental data are then used to develop equations of state, which efficiently embody the gained knowledge of the behavior of the fluid substance, allow for interpolation and, to some extent, extrapolation. However, the acquisition of low-uncertainty experimental data, and thus the development of accurate equations of state, is often time-consuming and expensive. For substances for which suitable force field models exist, molecular modeling and simulation are well-suited to generate thermodynamic data or to augment experimental data, however, at the expense of larger uncertainties. The major goal of this work is to present a new approach for the development of equations of state using (1) symbolic regression, which is a machine learning based model development approach, (2) optimal experimental design, and (3) efficient data acquisition. We demonstrate this approach using the example of density data of an air-like binary mixture (0.2094,hbox {O}_{2},+,0.7906,hbox {N}_{2}) over the temperature range from {100},{textrm{K}} to {300},{textrm{K}} at pressures of up to {8},{textrm{MPa}}, which covers the gaseous, liquid, and supercritical regions. For this purpose, an experimental data set published by von Preetzmann et al. (Int. J. Thermophys. 42, 2021) and molecular simulation data sampled in this work are used. The two data sets are compared in terms of acquisition time, cost, and uncertainty, showing that an optimized combination of experimental and simulation data leads to lower cost while maintaining low uncertainties.

This paper deals with the development of a computational code for the numerical simulation of wave propagation through domains with a complex geometry consisting in both solids and moving fluids. The emphasis is on the numerical simulation of ultrasonic flowmeters (UFMs) by modeling the wave propagation in solids with the equations of linear elasticity (ELE) and in fluids with the linearized Euler equations (LEEs). This approach requires high performance computing because of the high number of degrees of freedom and the long propagation distances. Therefore, the numerical method should be chosen with care. In order to minimize the numerical dissipation which may occur in this kind of configuration, the numerical method employed here is the nodal discontinuous Galerkin (DG) method. Also, this method is well suited for parallel computing. To speed up the code, almost all the computational stages have been implemented to run on graphical processing unit (GPU) by using the compute unified device architecture (CUDA) programming model from NVIDIA. This approach has been validated and then used for the two-dimensional simulation of gas UFMs. The large contrast of acoustic impedance characteristic to gas UFMs makes their simulation a real challenge.

SUMMARYWe present an operator‐splitting scheme for fluid–structure interaction (FSI) problems in hemodynamics, where the thickness of the structural wall is comparable to the radius of the cylindrical fluid domain. The equations of linear elasticity are used to model the structure, while the Navier–Stokes equations for an incompressible viscous fluid are used to model the fluid. The operator‐splitting scheme, based on the Lie splitting, separates the elastodynamics structure problem from a fluid problem in which structure inertia is included to achieve unconditional stability. We prove energy estimates associated with unconditional stability of this modular scheme for the full nonlinear FSI problem defined on a moving domain, without requiring any sub‐iterations within time steps. Two numerical examples are presented, showing excellent agreement with the results of monolithic schemes. First‐order convergence in time is shown numerically. Modularity, unconditional stability without temporal sub‐iterations, and simple implementation are the features that make this operator‐splitting scheme particularly appealing for multi‐physics problems involving FSI. Copyright © 2013 John Wiley & Sons, Ltd.

Adaptive r and h — r algorithms for boundary elements

Iterative methods are considered for a class of saddle point problems with a penalty term arising from finite element discretizations of certain elliptic problems. An optimal preconditioner which is independent of the discretization and the penalty parameter is constructed. This approach is then used to design an iterative method with a convergence rate bounded independently of the Lamé parameters occurring in the equations of linear elasticity.

A fluid pressure and deformation analysis for geological sequestration of carbon dioxide

In this work, the different kinds of fluid flow interactions with piezo smart materials have been discussed for energy harvesting. The present work has been classified into the following categories: (i) experimental investigations (ii) simulation and (iii) mathematical modelling. In section (i) different experimental set-ups such as harvesting of energy with the help of vortex flow, turbulent flow, cross flow, flow in an open channel and closed channel and flow through nozzles have been examined. In section (ii) simulations studies performed with different tools/software like ANSYS/fluent, COSMOL etc. have been detailed. Lastly, in section (iii) different mathematical equations such as Navier-Stokes equation of motion, Continuity equation, finite element method, numerical methods, transport equations, Bernoulli equation, equation of linear elasticity, fluid structural equations, piezoelectric equations and coupled-wave equations are described for generation of energy with fluid’s interaction. The present work has to fulfil two aims: (i) active engineers can choose the best appropriate methodology for their work in the same field (ii) researchers can know about how the area of energy harvesting has been grown in various decades, what are the practical application of this field with real life and the literature gaps of the field.

A discontinuous Galerkin approach was presented for the numerical simulation of acoustic waves propagation in transit-time ultrasonic flowmeters. The wave propagation in the fluid part was described by the linearized Euler equations and the equations of linear elasticity was adopted in the solid part. Stationary background flow field of ultrasonic propagation was calculated by solving the k-e turbulent flow model. By linearizing the acoustic variables, the governing equation of linear sound propagation in the adiabatic state was obtained. By defining the normal velocity on the surface of transducers, the average sound pressure on the surface of receiver under downstream and upstream propagation could be generated. The simulated time difference was smaller than that of the experimental results when comparing the average sound pressure for downstream and upstream propagation. But the error of time difference between the experimental and simulation was within 5%.

The problem of the compliant surface journal bearing is investigated. The mathematical problem is divided into two parts, the elasticity problem and the fluid problem. The equations of linear elasticity, which represent the elasticity problem, are solved; (a) by expansion of the solution in eigenfunctions (b) by the Raleigh Ritz method, and (c) by a collocation method. Reynolds’ equation, which represents the fluid problem, is then coupled with the solution to the elasticity problem. The corresponding finite difference representation of Reynolds’ equation, which is a set of fourth-order nonlinear algebraic equations, is inverted by Newton iteration. Finally, numerical results are presented.