A Variational Approach for the Navier–Stokes System

We introduce a variational approach to treat the regularity of the Navier–Stokes equations both in dimensions 2 and 3. Though the method allows the full treatment in dimension 2, we seek to precisely stress where it breaks down for dimension 3. The basic feature of the procedure is to look directly for strong solutions, by minimizing a suitable error functional that measures the departure of feasible fields from being a solution of the problem. By considering the divergence-free property as part of feasibility, we are able to avoid the explicit analysis of the pressure. Two main points in our analysis are:

A Method for Determining Model‐Structure Errors and for Locating Damage in Vibrating Systems

A method is proposed for the determination of natural frequencies and mode shapes of a system which is constrained so that unknown stiffnesses are replaced by rigid connections. The constraint is not imposed physically but only in mathematics so that the behaviour of the constrained system is inferred from the unconstrained measurements. Since stiffnesses which are made rigid cannot experience any elastic strain they can have no effect on the inferred measurements. A procedure for comparing the inferred measurements with similarly constrained finite element predictions can be used to determine modelstructure errors. Damage, such as a crack in a beam, can be located by comparing the inferred measurements from the structure in its undamaged and current states. It is demonstrated how unmeasured rotations may be constrained by using rigidbody modes and a reduction/expansion transformation from a finite element model.Sommario. Viene proposto un metodo per la determinazione delle frequenze proprie e dei modi di vibrazione di un sistema vincolato in modo tale che alcuni elementi elastici siano sostituiti da collegamenti rigidi. Il vincolo non viene imposto fisicamente, ma solo matematicamente, e pertanto il comportamento del sistema vincolato viene dedotto dalle misure sul sistema non vincolato. Poiché gli elementi che sono resi rigidi non possono subire alcuna deformazione elastica, essi non hanno certamente alcun effetto sulle misure dedotte per il sistema vincolato. Una procedura che mette a confronto le misure dedotte per il sistema vincolato con le previsioni fornite da un modello ad elementi finiti con analoghi vincoli, può essere utilizzata per determinare errori nella struttura del modello. Danni del tipo di una cricca su una trave possono essere localizzati confrontando le misure dedotte – per sistemi analogamente vincolati – da quelle effettuate sulla struttura non danneggiata e sulla struttura danneggiata. Si dimostra come si possono imporre vincoli sulle rotazioni (non misurate) utilizzando i modi di corpo rigido dell'elemento e una tecnica di riduzione/espansione dei gradi di libertà di un modello ad elementi finiti.     

A Variational Approximation Scheme for¶Three-Dimensional Elastodynamics¶with Polyconvex Energy

We construct a variational approximation scheme for the equations of three-dimensional elastodynamics with polyconvex stored energy. The scheme is motivated by some recently discovered geometric identities (Qin [18]) for the null Lagrangians (the determinant and cofactor matrix), and by an associated embedding of the equations of elastodynamics into an enlarged system which is endowed with a convex entropy. The scheme decreases the energy, and its solvability is reduced to the solution of a constrained convex minimization problem. We prove that the approximating process admits regular weak solutions, which in the limit produce a measure-valued solution for polyconvex elastodynamics that satisfies the classical weak form of the geometric identities. This latter property is related to the weak continuity properties of minors of Jacobian matrices, here exploited in a time-dependent setting. Accepted November 18, 2000?Published online April 23, 2001     

Metastability and Dynamics of Discrete Topological Singularities in Two Dimensions: A Γ-Convergence Approach

This paper aims at building a variational approach to the dynamics of discrete topological singularities in two dimensions, based on Γ-convergence. We consider discrete systems, described by scalar functions defined on a square lattice and governed by periodic interaction potentials. Our main motivation comes from XY spin systems, described by the phase parameter, and screw dislocations, described by the displacement function. For these systems, we introduce a discrete notion of vorticity. As the lattice spacing tends to zero we derive the first order Γ-limit of the free energy which is referred to as renormalized energy and describes the interaction of vortices. As a byproduct of this analysis, we show that such systems exhibit increasingly many metastable configurations of singularities. Therefore, we propose a variational approach to the depinning and dynamics of discrete vortices, based on minimizing movements. We show that, letting first the lattice spacing and then the time step of the minimizing movements tend to zero, the vortices move according with the gradient flow of the renormalized energy, as in the continuous Ginzburg–Landau framework.     

A Variational Framework for Spectral Approximations of Kohn–Sham Density Functional Theory

We reformulate the Kohn–Sham density functional theory (KSDFT) as a nested variational problem in the one-particle density operator, the electrostatic potential and a field dual to the electron density. The corresponding functional is linear in the density operator and thus amenable to spectral representation. Based on this reformulation, we introduce a new approximation scheme, termed spectral binning, which does not require smoothing of the occupancy function and thus applies at arbitrarily low temperatures. We prove convergence of the approximate solutions with respect to spectral binning and with respect to an additional spatial discretization of the domain.     

A Γ-Convergence Approach to Stability of Unilateral Minimality Properties in Fracture Mechanics and Applications

We prove a stability result for a large class of unilateral minimality properties which arise naturally in the theory of crack propagation proposed by Francfort & Marigo in [14]. Then we give an application to the quasistatic evolution of cracks in composite materials. The main tool in the analysis is a Γ-convergence result for energies of the form where S(u) is the jump set of u and is a sequence of rectifiable sets with We prove that no interaction occurs in the Γ-limit process between the bulk and the surface part of the energy. Relying on this result, we introduce a new notion of convergence for (N−1)-rectifiable sets called σ-convergence, which is useful in the study of the stability of unilateral minimality properties.     

A N‐dynamics Model for Predicting N‐behavior Subject to Environmentally Friendly Fertilization Practices: II –Numerical Model and Model Validation

Nitrogen dynamics in the soil under the condition of environmentally friendly fertilization practices (EFFPs) is described by a comprehensive Ndynamics model. The model (first paper of this series, Transport in Porous Media 31(3) (1998), 249–274) is different from other models in its capability of simulating the special phenomena related to the application of EFFPs. In this paper, a finite difference method is used to solve the mathematical model. The numerical model is verified by simulating several water flow and conservative solute transport problems with existing numerical or analytic solutions. The good agreements between our simulation results and the solutions given by others show that our model is reliable in simulating flow and transport problems in the soil. Preliminary model validation is conducted by applying the model to simulate two field experiments. The acceptable agreements between our numerical simulation results and experimental data demonstrate that the model can reasonably model Ndynamics in the soil under field conditions.     

A Charming Science Because of Its Complexity and Usefulness

I like the cover page stories of China Particuology,which teach me a lot of history of the science and technology of particle processing. From winnowing and grain processing, to ore upgrading and gunpowder manufacture, and to Pythagoras‘ theorem, we learned sizing,flow, structure, and function, and the active invention ofliving techniques to logical thinking of fundamental relationships behind the techniques, i.e., science.     

A Variational Framework for the Thermomechanics of Gradient-Extended Dissipative Solids – with Applications to Diffusion,Damage and Plasticity

Journal of Elasticity - The paper presents a versatile framework for solids which undergo nonisothermal processes with irreversibly changing microstructure at large strains. It outlines rate-type...     

A Structural Model of Ceramic Deformation and Fracture∗

ABSTRACT

A new structural model of deformation and fracture of ceramics is presented. The real material is simulated as a periodic grid of hexagonal elastic grains connected with elastic bondings. A method for deformation analysis of such a medium is introduced, and the effective modules of the material are calculated. The solution of the problem of wave propagation gives dispersion correlations. The deformation model introduced and the procedure of Fourier transforms yields a Green function for an infinite periodic grid of nondeformable hexagonal grains that are connected with elastic bondings. The fundamental solution is used to examine the strength of the medium with local defects and to compare it with the strength of a defectless material.     

Multi-scale Fractured Coal Gas–Solid Coupling Model and Its Applications in Engineering Projects

With coal mining entering the geological environment of “high stress, rich gas, strong adsorption and low permeability,” the difficulty of joint coal and gas extraction clearly augments, the risk of solid–gas coupling dynamic disasters greatly increases, and the underlying mechanisms become more complex. In this paper, based on the characteristics of coal’s multi-scale structure and spatiotemporal variation, the multi-scale fractured coal gas–solid coupling model (MSFM) was built. In this model, the interaction between coal matrix and its fractures and the mechanical characteristics of gas-bearing coal were considered, as well as their coupling relationship. By MATLAB software, the stress–damage–seepage numerical computation programs were developed, which were applied into Comsol Multiphysics to simulate gas flow caused by coal mining. The simulation results showed the spatial variability of coal elastic modulus and cross-flow behaviors of coal seam gas, which were superior to the results of traditional gas–solid coupling model. And the numerical results obtained from MSFM were closer to the measured results in field, while the computation results of traditional model were slightly higher than the measured results. Furthermore, the MSFM in a large scale was verified by field engineering project.     

A Representation Theorem for Material Tensors of?Weakly-Textured Polycrystals and?Its Applications in?Elasticity

Material tensors pertaining to polycrystalline aggregates should manifest also the influence of crystallographic texture on the material properties in question. In this paper we make use of tensors which form bases of irreducible representations of the rotation group and prove a representation theorem by which a given material tensor of a weakly-textured polycrystal is expressed as a linear combination of an orthonormal set of irreducible basis tensors, with the components given explicitly in terms of texture coefficients and a set of undetermined material parameters. Once the irreducible basis tensors that appear in the formula are determined, the representation formula, which is valid for all texture and crystal symmetries, will delineate quantitatively the effect of crystallographic texture on the material tensor in question. We present an integral formula and an orthonormalization process which serve as the basis for a procedure to determine explicitly the irreducible basis tensors required in the representation formula. For applications we determine a set of irreducible basis tensors for the elasticity tensor and a set for fourth-order tensors that define constitutive equations in incompressible elasticity and Hill’s quadratic yield functions in plasticity. We show that orientation averaging of a tensor can be done easily if we have in hand a set of irreducible basis tensors for the decomposition of the tensor in question. As illustration we derive a formula, which is valid for all texture and crystal symmetries, for the elasticity tensor under the Voigt model.     

A Viscous Augmented Born–Infeld Model for Magneto hydrodynamic Flows

We establish a weak stability result for a viscous variant of the Augmented Born–Infeld equations. This model may be seen as a magnetic model for heat conducting viscous fluids interacting with a magnetic field. It includes a well-known shallow-water magnetohydrodynamic system.     

The Hohenheim Tyre Model: A validated approach for the simulation of high volume tyres – Part I: Model structure and parameterisation

The applied tyre model influences significantly the accuracy of vehicle simulations. This is especially the case for farm machinery that is equipped with high volume tyres and mostly suspended on one axle only. In order to account for the special properties of these tyres – such as the nonlinearities that come along with high deflections – a new tyre model was developed at the University of Hohenheim. During the development phase the main requirements to fulfil were short computation times, an easy to apply parameterisation process and a high model quality. In order to attain these goals an all new multi-spoke tyre model was developed. Various adaptations were made to the model structure in order to achieve a real-time factor of 0.6. All eighteen parameters have a physical meaning and can be determined with two in-house tyre test stands. Validation comprises aspects relevant to both handling and ride quality and will be addressed in part two of this publication series.     

Propagation Phenomena for A Reaction–Advection–Diffusion Competition Model in A Periodic Habitat

This paper is devoted to the study of propagation phenomena for a Lotka–Volterra reaction–advection–diffusion competition model in a periodic habitat. We first investigate the global attractivity of a semi-trivial steady state for the periodic initial value problem. Then we establish the existence of the rightward spreading speed and its coincidence with the minimal wave speed for spatially periodic rightward traveling waves. We also obtain a set of sufficient conditions for the rightward spreading speed to be linearly determinate. Finally, we apply the obtained results to a prototypical reaction–diffusion model.     

Stationary Shear Flows of Nematic Liquid Crystals: A Comprehensive Study via Ericksen–Leslie Model

Journal of Dynamics and Differential Equations - We apply the Ericksen–Leslie dynamic model to investigate stationary solutions of planar shear flows for nematic liquid crystals. Nematic...     

An Analytical Model of Porosity–Permeability for Porous and Fractured Media

The classic Kozeny–Carman equation (KC) uses parameters that are empirically based or not readily measureable for predicting the permeability of unfractured porous media. Numerous published KC modifications share this disadvantage, which potentially limits the range of conditions under which the equations are applicable. It is not straightforward to formulate non-empirical general approaches due to the challenges of representing complex pore and fracture networks. Fractal-based expressions are increasingly popular in this regard, but have not yet been applied accurately and without empirical constants to estimating rock permeability. This study introduces a general non-empirical analytical KC-type expression for predicting matrix and fracture permeability during single-phase flow. It uses fractal methods to characterize geometric factors such as pore connectivity, non-uniform grain or crystal size distribution, pore arrangement, and fracture distribution in relation to pore distribution. Advances include (i) modification of the fractal approach used by Yu and coworkers for industrial applications to formulate KC-type expressions that are consistent with pore size observations on rocks. (ii) Consideration of cross-flow between pores that adhere to a fractal size distribution. (iii) Extension of the classic KC equation to fractured media absent empirical constants, a particular contribution of the study. Predictions based on the novel expression correspond well to measured matrix and fracture permeability data from natural sandstone and carbonate rocks, although the currently available dataset for fractures is sparse. The correspondence between model calculation results and matrix data is better than for existing models.     

Darcy–Carreau Model and Nonlinear Natural Convection for Pseudoplastic and Dilatant Fluids in Porous Media

Transport in Porous Media - The linear and weakly nonlinear stability analyses are carried out to study instabilities in Darcy–Bénard convection for non-Newtonian inelastic fluids. The...     

A Novel Relative Permeability Model Based on Mixture Theory Approach Accounting for Solid–Fluid and Fluid–Fluid Interactions

A novel model is presented for estimating steady-state co- and counter-current relative permeabilities analytically derived from macroscopic momentum equations originating from mixture theory accounting for fluid–fluid (momentum transfer) and solid–fluid interactions (friction). The full model is developed in two stages: first as a general model based on a two-fluid Stokes formulation and second with further specification of solid–fluid and fluid–fluid interaction terms referred to as \(R_{{i}}\) (i =  water, oil) and R, respectively, for developing analytical expressions for generalized relative permeability functions. The analytical expressions give a direct link between experimental observable quantities (end point and shape of the relative permeability curves) versus water saturation and model input variables (fluid viscosities, solid–fluid/fluid–fluid interactions strength and water and oil saturation exponents). The general two-phase model is obeying Onsager’s reciprocal law stating that the cross-mobility terms \(\lambda _\mathrm{wo}\) and \(\lambda _\mathrm{ow}\) are equal (requires the fluid–fluid interaction term R to be symmetrical with respect to momentum transfer). The fully developed model is further tested by comparing its predictions with experimental data for co- and counter-current relative permeabilities. Experimental data indicate that counter-current relative permeabilities are significantly lower than corresponding co-current curves which is captured well by the proposed model. Fluid–fluid interaction will impact the shape of the relative permeabilities. In particular, the model shows that an inflection point can occur on the relative permeability curve when the fluid–fluid interaction coefficient \(I>0\) which is not captured by standard Corey formulation. Further, the model predicts that fluid–fluid interaction can affect the relative permeability end points. The model is also accounting for the observed experimental behavior that the water-to-oil relative permeability ratio \(\hat{{k}}_{\mathrm{rw}} /\hat{{\mathrm{k}}}_{\mathrm{ro}} \) is decreasing for increasing oil-to-water viscosity ratio. Hence, the fully developed model looks like a promising tool for analyzing, understanding and interpretation of relative permeability data in terms of the physical processes involved through the solid–fluid interaction terms \(R_{{i}}\) and the fluid–fluid interaction term R.