Elements of Continuum Mechanics and Conservation Laws

The Elastodynamic Finite Integration Technique (EFIT), originally developed by Fellinger et al.,1–3 represents a stable and efficient numerical code to model elastic wave propagation in linearly-elastic isotropic and anisotropic, homogeneous and heterogeneous as well as dissipative and nondissipative media. In previous works, the FIT discretization of the basic equations of linear elasticity, Hooke's law and Cauchy's equation of motion, was exclusively carried out in Cartesian coordinates. For problems in cylindrical geometries it is more suitable to use cylindrical coordinates. By that, axisymmetric problems can be treated in a two-dimensional staggered grid in the r,z-plane. The paper presents an EFIT version for axisymmetric problems in cylindrical coordinates called Cylindrical EFIT (CEFIT). After demonstrating the accuracy of the numerical code by a comparison between simulation results and analytical solutions, different examples of application are given. These examples include modeling of sound fields of ultrasonic transducers, thermoelastic laser sources, geophysical borehole probes, impact-echo measurements in layered media, and load simulations of the European Spallation Source (ESS) mercury target.

Reverse time migration (RTM) exhibits great advantages over other imaging methods because it is based on computing numerical solutions to a two-way wave equation. It does not suffer from dip limitation like one-way downward continuation techniques do, thus enabling overturned reflections to be imaged. As well as correctly handling multipathing, RTM has the potential to image internal multiples when the boundaries responsible for generating the multiples are present in the model. In isotropic media, one can use a scalar acoustic wave equation for RTM of pressure data. In anisotropic media, P- and SV-waves are coupled together so, formally, elastic wave equations must be used for RTM. A new wave equation for P-waves is proposed in tilted transversely isotropic (TTI) media that can be solved as part of an acoustic anisotropic RTM algorithm, using standard explicit finite differencing. If the shear velocity along the axis of symmetry is set to zero, stable numerical solutions can be computed for media with a vertical axis of symmetry and ε not less than δ. In TTI media with rapid variations in the direction of the axis of symmetry, setting the shear velocity along the axis of symmetry to zero can cause numerical solutions to become unstable. A solution to this problem is proposed that involves using a small amount of nonzero shear velocity. The amount of shear velocity added is chosen to remove triplications from the SV wavefront and to minimize the anisotropic term of the SV reflection coefficient. We show modeling and high-quality RTM results in complex TTI media using this equation.

Surface reflection tomography is an inversion method that attempts to determine simultaneously the subsurface elastic parameters and reflector depths from pre-stack seismic data. To date, this technique has been applied to acoustic and isotropic elastic media. In this paper, we study surface reflection tomography in the anisotropic elastic case. Anisotropic media are described by many more elastic parameters than are needed to describe isotropic media; these additional parameters are useful for obtaining rock properties such as crack density, pore shape and fracture strike that cannot be found with isotropic methods. Similar to isotropic surface reflection tomography, many features of an anisotropic model are poorly resolved due to limited ray path coverage; the vertical smearing of isotropic tomography occurs in each elastic parameter in the anisotropic problem. Unlike the isotropic problem, however, there is additional indeterminacy in the solution of the anisotropic tomography problem because of ambiguity amongst the several elastic parameters needed to describe anisotropic media. Also unlike the isotropic problem, the reflector depths participate very strongly in this ambiguity. We investigate the nature of this indeterminacy by studying the null space for linearized tomography, that is, the class of model perturbations of a background medium which, to first order, cause no perturbation at all in the surface reflection traveltimes. Such model perturbations cannot be determined from the traveltime perturbations; describing these null space model perturbations gives insight into the indeterminacy in the anisotropic problem. Complementary to computational approaches towards identifying the null space for discrete formulations of tomography, we study a continuum formulation. Our first set of results concerns the ‘elastic’ null space: perturbations of the elastic parameters (with zero depth perturbation) which cause no perturbation in the traveltimes. The results here are very similar to previous results for cross-well transmission tomography. As expected, the ‘elastic’ null space is larger than the isotropic null space due to the ambiguity amongst the elastic parameters. We identify three categories of model perturbations in the ‘elastic’ null space. The first category consists of model perturbations for which the perturbation in each of the individual elastic parameters is itself either in the isotropic null space, or in a closely related set of model perturbations which we call the odd isotropic null space. Elements in the second category are anisotropic versions of the most well-known isotropic null space elements: perturbations which are polynomials in the horizontal variable with coefficients which are functions of the depth variable satisfying certain linear integral constraints; unlike the isotropic problem, the integral constraints in the anisotropic problem couple together the several elastic parameters. The third category consists of model perturbations satisfying zero boundary conditions at the surface and at the reflector for which a specific linear combination of integrals and derivatives of the several elastic parameters is in the isotropic null space. In particular, there are model perturbations in this third category which represent anomalies that are completely contained in the interior of the model and yet are in the ‘elastic’ null space; this behaviour is different from the isotropic problem. These categories are sufficient to describe the ‘elastic’ null space completely. We demonstrate that every model perturbation in the ‘elastic’ null space is the sum of an element in the first category (indicating an indeterminacy of the same nature as in the isotropic problem in each of the elastic parameters separately) and an element in the third category (indicating an ambiguity amongst the parameters). The second category gives a rich family of examples of sums of null space elements in the first and third categories, and thereby gives a sense of just how large the ‘elastic’ null space is. Moreover, we show that the traveltime perturbations caused by an elastic perturbation determine only a small number of features of the elastic perturbation which distinguish between the several elastic parameters. We identify these features pre-cisely: they are functions of the horizontal variable representing vertical averages of combinations of the elastic parameters and their derivatives. Elastic parameters influencing the vertical velocity appear more prominently in these features than those influencing the horizontal velocity, and these features are closely related to the zero-offset traveltimes and the normal moveout velocities. Our second set of results concerns the so-called velocity–depth ambiguity, more precisely here the elastic parameter versus the depth parameter ambiguity. We demonstrate that for any smooth depth perturbation, there is a perturbation in the elastic parameters for which the combined elastic and depth perturbations cause zero traveltime perturbations; moreover, this can be accomplished by an elliptically anisotropic elastic perturbation with a vertical axis of symmetry, perturbing the vertical velocity alone. This implies that the velocity–depth ambiguity cannot be resolved here as it can be in the isotropic case. We show an example of an anticlinal structure embedded in a homogeneous background whose reflection traveltimes exactly match those from a model with elliptically anisotropic elastic perturbations overlying a flat reflector. We also draw a number of conclusions from these results for the special case of transversely isotropic media with a vertical symmetry axis.

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

The Lie group method is a powerful technique for obtaining analytical solutions for various nonlinear differential equations. This study aimed to explore the behavior of nonlinear elastic wave equations and their underlying physical properties using Lie group invariants. We derived eight-dimensional symmetry algebra for the (3+1)-dimensional nonlinear elastic wave equation, which was used to obtain the optimal system. Group-invariant solutions were obtained using this optimal system. The same analysis was conducted for the damped version of this equation. For the conservation laws, we applied Noether’s theorem to the nonlinear elastic wave equations owing to the availability of a classical Lagrangian. However, for the damped version, we cannot obtain a classical Lagrangian, which makes Noether’s theorem inapplicable. Instead, we used an extended approach based on the concept of a partial Lagrangian to uncover conservation laws. Both techniques account for the conservation laws of linear momentum and energy within the model. These novel approaches add an application of variational calculus to the existing literature. This offers valuable insights and potential avenues for further exploration of the elastic wave equations.

This article provides the application of the high-order, staggered-grid, finite-difference scheme to model elastic wave propagation in 3-D isotropic media. Here, we use second-order, temporal- and high-order spatial finite-difference formulations with a staggered grid for discretization of the 3-D elastic wave equations of motion. The set of absorbing boundary conditions based on paraxial approximations of 3-D elastic wave equations are applied to the numerical boundaries. The trial results for the salt model show that the numerical dispersion is decreased to a minimum extent, the accuracy high and diffracted waves abundant. It also shows that this method can be used for modeling wave propagation in complex media with the lateral variation of velocity.

The present paper is concerned the effects of thermo elastic waves in a composite multilayered hollow pipes which contain inner and outer zinc layers bonded by Linear Elastic Material with Voids (LEMV) is considered. The equation of movement is derived by means of the constitutive equations of linear thermo elasticity. The equation of movement and heat conduction models are decoupled by the displacement of potentials which are constructed based on equilibrium equations of elasticity. The dispersion equation are acquired by means of traction free boundary conditions and are numerically analyzed for Zinc material. The enumerated frequency, phase velocity are presented graphically for LEMV and CFRP layers.

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.

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.

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

Computational modeling is a key resource to gather insight into physical systems in modern scientific research and engineering. While access to large amount of data has fueled the use of machine learning to recover physical models from experiments and increase the accuracy of physical simulations, purely data-driven models have limited generalization and interpretability. To overcome these limitations, we propose a framework that combines symbolic regression (SR) and discrete exterior calculus (DEC) for the automated discovery of physical models starting from experimental data. Since these models consist of mathematical expressions, they are interpretable and amenable to analysis, and the use of a natural, general-purpose discrete mathematical language for physics favors generalization with limited input data. Importantly, DEC provides building blocks for the discrete analog of field theories, which are beyond the state-of-the-art applications of SR to physical problems. Further, we show that DEC allows to implement a strongly-typed SR procedure that guarantees the mathematical consistency of the recovered models and reduces the search space of symbolic expressions. Finally, we prove the effectiveness of our methodology by re-discovering three models of continuum physics from synthetic experimental data: Poisson equation, the Euler’s elastica and the equations of linear elasticity. Thanks to their general-purpose nature, the methods developed in this paper may be applied to diverse contexts of physical modeling.

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.