Balance Laws Research Articles

An Approach to Growth Mechanics Based on the Analytical Mechanics of Nonholonomic Systems

Motivated by the convenience, in some biomechanical problems, of interpreting the mass balance law of a growing medium as a nonholonomic constraint on the time rate of a structural descriptor known as growth tensor, we employ some results of analytical mechanics to show that such constraint can be studied variationally. Our purpose is to move a step forward in the formulation of a field theory of the mechanics of volumetric growth by defining a Lagrangian function that incorporates the nonholonomic constraint of the mass balance. The knowledge of such Lagrangian function permits, on the one hand, to deduce the dynamic equations of a growing medium as the result of a variational procedure known as Hamilton–Suslov Principle (clearly, up to non-potential generalized forces that are accounted for by extending this procedure), and, on the other hand, to study the symmetries and conservation laws that pertain to a given growth problem. While this second issue is not investigated in this work, we focus on the first one, and we show that the Euler–Lagrange equations of the considered growing medium, which describe both its motion and the evolution of the growth tensor, can be obtained by reformulating a variational method developed by other authors. We discuss the main features of this method in the context of growth mechanics, and we show how our procedure is able to improve them.

IGA Approximations of Elastic Interfaces and their Defects in an Elastic Medium with Couple Stress

The objective of this work is to develop and implement a computational algorithm for calculating stress and couple-stress fields induced by bulk and interfacial line defects such as dislocations and generalized disclinations within phase/grain boundaries. The thermodynamic driving forces on line and planar defects, responsible for coupled plasticity and interface motion, fully coupled to stress, couple-stress and applied boundary conditions are also computed. A continuum approach for small deformations is considered following the formulation outlined in Acharya and Fressengeas (Continuum mechanics of the interaction of phase boundaries and dislocations in solids. In: Differential Geometry and Continuum Mechanics, pages 123–165, 2015), extended herein in the thermodynamics to accommodate physically necessary ingredients that arose in the modeling in Zhang and Acharya (J Mech Phys Solids 119:188–223, 2018), Zhang et al. (J Mech Phys Solids 114:258–302, 2018). Constitutive relations are derived from kinematics, balance laws and from the use of the second law of thermodynamics in global form. One of the challenges presented by this approach is the inclusion of couple stresses, Toupin (Arch Rational Mech Anal 17(2):85–112, 1964), and the consequent treatment of 4th order systems arising from the equations of balances of linear and angular momentum. In order to deal with these equations, the classical FEM approach is replaced by iso-geometric analysis (IGA), as proposed by Hughes et al. (Comput Methods Appl Mech Eng 194(39):4135–4195, 2005). Results on stress and couple stress fields of the various defects involved are computed. Further, the coupling of the dislocation density with the eigenwall allows for the capturing of the shear parallel to the grain boundary, which has been observed experimentally and through molecular dynamics.

Dynamical behaviour of a logarithmically sensitive chemotaxis model under time-dependent boundary conditions

Abstract This article studies the dynamical behaviour of classical solutions of a hyperbolic system of balance laws, derived from a chemotaxis model with logarithmic sensitivity, with time-dependent boundary conditions. It is shown that under suitable assumptions on the boundary data, solutions starting in the $H^2$ -space exist globally in time and the differences between the solutions and their corresponding boundary data converge to zero as time goes to infinity. There is no smallness restriction on the magnitude of the initial perturbations. Moreover, numerical simulations show that the assumptions on the boundary data are necessary for the above-mentioned results to hold true. In addition, numerical results indicate that the solutions converge asymptotically to time-periodic states if the boundary data are time-periodic.

Mathematical Challenges for the Theory of Hyperbolic Balance Laws in Fluid Mechanics: Complexity, Scales, Randomness

Understanding the dynamics of hyperbolic balance laws is of paramount interest in the realm of fluid mechanics. Nevertheless, fundamental questions on the analysis and the numerics for distinctive hyperbolic features related to turbulent flow motion remain vastly open. Recent progress on the mathematical side reveals novel routes to face these concerns. This includes findings about the failure of the entropy principle to ensure uniqueness, the use of structure-preserving concepts in high-order numerical methods, and the advent of tailored probabilistic approaches. Whereas each of these three directions on hyperbolic modelling are of completely different origin they are all linked to small- or subscale features in the solutions which are either enhanced or depleted by the hyperbolic nonlinearity. Thus, any progress in the field might contribute to a deeper understanding of turbulent flow motion on the basis of the continuum-scale mathematical models. We present an overview on the mathematical state-of-the-art in the field and relate it to the scientific work in the DFG Priority Research Programme 2410. As such, the survey is not necessarily targeting at readers with comprehensive knowledge on hyperbolic balance laws but instead aims at a general audience of reseachers which are interested to gain an overview on the field and associated challenges in fluid mechanics.

Efficient numerical approximations for a nonconservative nonlinear Schrödinger equation appearing in wind‐forced ocean waves

AbstractWe consider a nonconservative nonlinear Schrödinger equation (NCNLS) with time‐dependent coefficients, inspired by a water waves problem. This problem does not have mass or energy conservation, but instead mass and energy change in time under explicit balance laws. In this paper, we extend to the particular NCNLS two numerical schemes which are known to conserve energy and mass in the discrete level for the cubic nonlinear Schrödinger equation. Both schemes are second‐order accurate in time, and we prove that their extensions satisfy discrete versions of the mass and energy balance laws for the NCNLS. The first scheme is a relaxation scheme that is linearly implicit. The other scheme is a modified Delfour–Fortin–Payre scheme, and it is fully implicit. Numerical results show that both schemes capture robustly the correct values of mass and energy, even in strongly nonconservative problems. We finally compare the two numerical schemes and discuss their performance.

A High-Order Well-Balanced Discontinuous Galerkin Method for Hyperbolic Balance Laws Based on the Gauss-Lobatto Quadrature Rules

In this paper, we develop a high-order well-balanced discontinuous Galerkin method for hyperbolic balance laws based on the Gauss-Lobatto quadrature rules. Important applications of the method include preserving the non-hydrostatic equilibria of shallow water equations with non-flat bottom topography and Euler equations in gravitational fields. The well-balanced property is achieved through two essential components. First, the source term is reformulated in a flux-gradient form in the local reference equilibrium state to mimic the true flux gradient in the balance laws. Consequently, the source term integral is discretized using the same approach as the flux integral at Gauss-Lobatto quadrature points, ensuring that the source term is exactly balanced by the flux in equilibrium states. Our method differs from existing well-balanced DG methods for shallow water equations with non-hydrostatic equilibria, particularly in the aspect that it does not require the decomposition of the source term integral. The effectiveness of our method is demonstrated through ample numerical tests.

Exact solution of the generalized Riemann problem for the Eulerian droplet model with particle pressure

In this paper, we investigate the generalized Riemann problem for the Eulerian droplet model with particle pressure. By introducing two new state variables, the generalized Riemann problem for a system of balance law (nonhomogeneous system) is neatly transferred into the Riemann problem for a system of conservation law (homogeneous system). Four different kinds of exact solutions are presented in fully explicit forms, which are given in terms of two nonconstant states separated by a shock or a generalized rarefaction wave or by a combination of such wave profiles.

Hyperbolic Balance Laws: Interplay between Scales and Randomness

Hyperbolic balance laws are fundamental in the mathematical modeling of transport-dominated processes in natural, socio-economic and engineering sciences. The aim of the workshop was to discuss open questions in the area of nonlinear hyperbolic conservation and balance laws. We have focused on a delicate interplay between scale hierarchies and random/stochastic effects and discuss them from analytical, numerical and modeling point of view. This leads to questions of admissibility criteria connecting to ill-posedness of weak entropy solutions, hyperbolic problems with non-local terms, mean field theory, multiscale and structure preserving numerical methods, random solutions and uncertainty quantification methods, as well as data-based methods.

Partially Dissipative Viscous System of Balance Laws and Application to Kuznetsov–Westervelt Equation

Partially Dissipative Viscous System of Balance Laws and Application to Kuznetsov–Westervelt Equation

The minimal control time for the exact controllability by internal controls of 1D linear hyperbolic balance laws

The minimal control time for the exact controllability by internal controls of 1D linear hyperbolic balance laws

Semi-implicit-Type Order-Adaptive CAT2 Schemes for Systems of Balance Laws with Relaxed Source Term

AbstractIn this paper, we present two semi-implicit-type second-order compact approximate Taylor (CAT2) numerical schemes and blend them with a local a posteriori multi-dimensional optimal order detection (MOOD) paradigm to solve hyperbolic systems of balance laws with relaxed source terms. The resulting scheme presents the high accuracy when applied to smooth solutions, essentially non-oscillatory behavior for irregular ones, and offers a nearly fail-safe property in terms of ensuring the positivity. The numerical results obtained from a variety of test cases, including smooth and non-smooth well-prepared and unprepared initial conditions, assessing the appropriate behavior of the semi-implicit-type second order CATMOOD schemes. These results have been compared in the accuracy and the efficiency with a second-order semi-implicit Runge-Kutta (RK) method.

GRP – A direct Godunov extension

The classical paper [40] by S. K. Godunov had a revolutionary effect on the field of numerical simulations of compressible fluid flows. Yet, for some twenty years after its publication, it was mostly used by the mathematical community in the Soviet Union. The seminal paper of van Leer [85] has inaugurated the period of universal interest in high-resolution extensions of Godunov's scheme. The first step to take was to modify the (locally) self-similar solution to the Riemann Problem (at discontinuities) by allowing piecewise-polynomial (rather than piecewise-constant) initial data. The GRP (Generalized Riemann Problem) analysis [6] provided analytical solutions (for piecewise-linear data) that could be readily implemented in a high-resolution robust code. This review paper focuses on the evolution of some mathematical aspects of this method. It addresses the three concepts of consistency, stability and convergence in the context of compact finite difference (or finite volume) schemes for systems of nonlinear hyperbolic conservation laws. The treatment utilizes the framework of “balance laws”, a common viewpoint in relevant physical conservation laws. The first significant observation is that under very mild conditions a weak solution is indeed a solution to the balance law (obtained by a formal application of the Gauss-Green formula). Since high-resolution schemes require the computation of several quantities per mesh cell (e.g., slopes), the notion of “consistency” must be extended to this framework. A combination of consistency hypothesis with stability of the scheme leads to a suitable convergence theorem, generalizing the classical convergence theorem of Lax and Wendroff [52]. Finally, the limit functions are shown to be entropy solutions by using a notion of “Godunov compatibility”, which serves as a substitute to the entropy condition.

Moving horizon estimation for pipeline leak detection, localization, and constrained size estimation

Advanced pipeline leak detection and localization techniques are needed to reduce greenhouse gas emissions from hydrocarbon transportation pipelines. Developing effective leak detection and localization methods is challenging due to the spatiotemporal dynamics of process variables, the presence of process/measurement disturbances and constraints, and the limited measurement data. To address this issue, this manuscript proposes a novel moving horizon estimation design for pipeline leak detection, constrained estimation of leak size and location by using an infinite-dimensional pipeline hydraulic model. Based on the mass and momentum balance laws and the Cayley–Tustin time-discretization method, an infinite-dimensional discrete-time pipeline hydraulic model is proposed considering (unknown but bounded) disturbance and leak. By introducing a coordinate transformation, we decouple the leak size and location estimation problems. The implementable discrete-time moving horizon estimator and observer are designed for constrained leak size and location estimation. The effectiveness of the proposed designs is validated via simulation examples.

A unified discontinuous Galerkin formulation for interfacial multiphysics modeling of thermo-chemically driven fracture

Many engineering and natural materials exhibit coupled thermo-chemo-mechanical phenomena, which can result in embrittlement and fracture. These fractures, in turn, can alter the subsequent thermal, chemical, and mechanical response. We present a theoretical formulation and computational framework for the analysis of thermo-chemically fractured solids, with emphasis on the post-fracture thermal and chemical interfacial behavior. The theoretical model is based on the thermodynamically-consistent formulation of Loeffel and Anand (IJP, 2011). The computational method extends the scalable discontinuous Galerkin/Cohesive Zone Model (DG/CZM) of Radovitzky et al. (CMAME, 2011) to thermo-chemo-mechanics, which facilitates coupled, large-scale simulations of materials and structures containing failed interfaces. In the proposed framework, all balance laws are enforced weakly via the DG formalism, resulting in a unified formulation for multiphysics problems in solids. This naturally enables the incorporation of general interface models, e.g. to account for effects such as the aeolotropic reduction in thermochemical transport due to the presence of fractures, or the acceleration of chemical reactions along crack flanks. The approach is verified against two analytical solutions of boundary value problems drawn from thermo-poro-elasticity and thermally-driven delamination. A scalable, three-dimensional simulation of thermochemically-driven concrete cracking illustrates the complete capabilities of the interfacial multiphysics modeling framework.

Stability of hybrid time integration scheme for Lord–Shulman thermopiezoelectricity

Based on the existing initial boundary value and variational problems of Lord–Shulman thermopiezoelectricity a transient analysis of the behavior of piezoelectric materials is performed numerically. For space discretization of the variational problem the finite element method is used and for time discretization a hybrid time integration scheme is constructed on the basis of Newmark scheme for hyperbolic equation and generalized trapezoidal rule for parabolic equations. The unconditional stability of the developed time integration scheme is proved for some specific values of the scheme parameters by making use of energy balance law for the obtained time-discretized variational problem. Finally, the applicability of the constructed numerical scheme is demonstrated by the results of the numerical experiment which are compared with the ones available in literature.

Regulations for Bat Protection in Mexico's Wind Farms

Wind energy development has expanded the fastest globally among all renewable sources during the last 20 years. However, wind farms have documented adverse impacts on bats, including mortality from collisions with turbine blades and disruptions to habitat and behavior. As the world's sixth most attractive economy for renewables, with 70 operating wind farms, Mexico and its bats now face escalating threats from the country's burgeoning wind industry. Despite this rapid growth, few studies have analyzed Mexico's regulatory framework to prevent, evaluate, and mitigate wind farm effects on bats. In this study, we reviewed Mexican laws and treaties that facilitate wind farm permitting, construction, operation, and decommissioning, and searched for guidelines that specifically address bat conservation. We found eight international pacts that promote wind power adoption along with three relevant articles in Mexico's Constitution. The General Law of Ecological Balance and Environmental Protection proved most pertinent for impact management. Supplementary guidelines from the Ministry of Environment and Natural Resources offer general strategies for evaluating wind farm impacts on bats, but adherence remains voluntary. Given expanding wind power investments across Mexico, we highlight the need for more stringent national standards that require preventative and corrective measures to protect bat populations. Tighter legislation and enforcement offer pathways toward environmentally sustainable wind energy development in Mexico.

A geometric formulation of Schaefer’s theory of Cosserat solids

The Cosserat solid is a theoretical model of a continuum whose elementary constituents are notional rigid bodies, having both positional and orientational degrees of freedom. In this article, we present a differential geometric formulation of the mechanics of a Cosserat solid, motivated by Schaefer’s “motor field” theory. The solid is modeled as a special principal fiber bundle (a Cartan space) and its configurations are related by bundle maps. We show that the classical Lagrangian strain measure of a Cosserat solid is the difference of two Cartan connections on the bundle. The infinitesimal strain is derived by a rigorous linearization and is revealed to be the Lie derivative of a Cartan connection along the vector field representing the infinitesimal deformation. Incompatibilities in a Cosserat solid are characterised by a non-flat Cartan connection whose curvature is recognized as the density of topological defects. Stresses are defined as vector bundle-valued differential forms that are work-dual to strains and balance laws in the limit of vanishing inertia are obtained via a d’Alembert principle. Constitutive equations, with an emphasis on recent applications to active oriented solids, are briefly discussed.

A Direct Entropic Approach to the Thermal Balance of Spontaneous Chemical Reactions.

When working with, and learning about, the thermal balance of a chemical reaction, we need to consider two overlapping but conceptually distinct aspects: one relates to the process of reallocating entropy between reactants and products (because of different specific entropies of the new substances compared to those of the old), and the other to dissipative processes. Together, they determine how much entropy is exchanged between the chemicals and their environment (i.e., in heating and cooling). By making explicit use of (a) the two conjugate pairs chemical amount (i.e., amount of substance) and chemical potential, and entropy and temperature, respectively, (b) the laws of balance of amount of substance on the one hand and entropy on the other, and (c) a generalized approach to the energy principle, it is possible to create both imaginative and formal conceptual tools for modeling thermal balances associated with chemical transformations in general and exothermic and endothermic reactions in particular. In this paper, we outline the concepts and relations needed for a direct approach to chemical and thermal dynamics, create a model of exothermic and endothermic reactions, including numerical examples, and discuss how to relate the direct entropic approach to traditional models of these phenomena.

An explicit well-balanced scheme on staggered grids for barotropic Euler equations

In this paper, we introduce a specific modification of the numerical fluxes in order to insure the well-balanced property of schemes on staggered grids for the Euler equations. This property is crucial for the numerical representation of equilibrium solutions of balance laws with source terms, like when describing flows subjected to gravity and a complex topography. We propose first and second order versions of the well-balanced scheme. The performances of the method are evaluated through a series of 1D and 2D benchmarks.

Dynamic large strain formulation for nematic liquid crystal elastomers

Liquid crystal elastomers (LCEs) are a class of materials which exhibit an anisotropic behavior in their nematic state due to the main orientation of their rod-like molecules called mesogens. The reorientation of mesogens leads to the well-known actuation properties of LCEs, i.e. exceptionally large deformations as a consequence of particular external stimuli, such as temperature increase. Another key feature of nematic LCEs is the capability to undergo deformation by constant stresses while being stretched in a direction perpendicular to the orientation of mesogens. During this plateau stage, the mesogens rotate towards the stretching direction. Such characteristic is defined as semisoft elastic response of nematic LCEs. We aim at modeling the semisoft behavior in a dynamic finite element method based on a variational-based mixed finite element formulation. The reorientation process of the rigid mesogens relative to the continuum rotation is introduced by micropolar drilling degrees of freedom. Responsible for the above-mentioned characteristics is an appropriate free energy function. Starting from an isothermal free energy function based on the small strain theory, we aim to widen it into the framework of large strains by identifying tensor invariants. In this work, we analyze the isothermal influence of the tensor invariants on the mechanical response of the finite element formulation and show that its space-time discretization preserves mechanical balance laws in the discrete setting.