The Fundamental Theorem of Surface Theory for Surfaces with Little Regularity

Consider a symmetric, positive definite matrix field of order two and a symmetric matrix field of order two that satisfy together the Gauss and Codazzi-Mainardi equations in a connected and simply connected open subset of R ². If the matrix fields are respectively of class C ² and C ¹, the fundamental theorem of surface theory asserts that there exists a surface immersed in the three-dimensional Euclidean space with these fields as its first and second fundamental forms. The purpose of this paper is to prove that this theorem still holds under the weaker regularity assumptions that the matrix fields are respectively of class W ^1,∞ _loc and L ^∞ _loc, the Gauss and Codazzi-Mainardi equations being then understood in a distributional sense.     

A New Approach to Front Propagation Problems: Theory and Applications

In this paper we present a new definition for the global in time propagation (motion) of fronts (hypersurfaces, boundaries) with a prescribed normal velocity, past the first time they develop singularities. We show that if this propagation satisfies a geometric maximum principle (inclusion‐avoidance)‐type property, then the normal velocity must depend only on the position of the front and its normal direction and principal curvatures. This new approach, which is more geometric and, as it turns out, equivalent to the level‐set method, is then used to develop a very general and simple method to rigorously validate the appearance of moving interfaces at the asymptotic limit of general evolving systems like interacting particles and reaction‐diffusion equations. We finally present a number of new asymptotic results. Among them are the asymptotics of (i) reaction‐diffusion equations with rapidly oscillating coefficients, (ii) fully nonlinear nonlocal (integral differential) equations and (iii) stochastic Ising models with long-range anisotropic interactions and general spin flip dynamics. (Accepted July 8, 1996)     

A Pedagogical Approach to the Thermodynamically Constrained Averaging Theory

The thermodynamically constrained averaging theory (TCAT) is an evolving approach for formulating macroscale models that are consistent with both microscale physics and thermodynamics. This consistency requires some mathematical complexity, which can be an impediment to understanding and efficient application of this model-building approach for the non-specialist. To aid understanding of the TCAT approach, a simplified model formulation approach is developed and used to show a more compact, but less general, formulation compared to the standard TCAT approach. This new simplified model formulation approach is applied to the case of binary species diffusion in a single-fluid-phase porous medium system, clearly showing a TCAT approach that is applicable to many other systems as well. Recent extensions to the TCAT approach that enable a priori parameter estimation, and approaches to leverage available TCAT modeling building results are also discussed.     

A New Approach to Equations with Memory

We discuss a novel approach to the mathematical analysis of equations with memory, based on a new notion of state. This is the initial configuration of the system at time t = 0 which can be unambiguously determined by the knowledge of the dynamics for positive times. As a model, for a nonincreasing convex function ${G : \mathbb{R}^+ \to \mathbb{R}^+}We discuss a novel approach to the mathematical analysis of equations with memory, based on a new notion of state. This is the initial configuration of the system at time t = 0 which can be unambiguously determined by the knowledge of the dynamics for positive times. As a model, for a nonincreasing convex function G : \mathbbR⁺ ? \mathbbR⁺{G : \mathbb{R}^+ \to \mathbb{R}^+} such that

$G(0) = \lim_{s\to 0}G(s) > \lim_{s\to\infty}G(s) >0 $G(0) = \lim_{s\to 0}G(s) > \lim_{s\to\infty}G(s) >0      5. Some Fundamental Aspects of Surface Modelling      A. Ian Murdoch 《Journal of Elasticity》2005,80(1-3):33-52 Attention is drawn to four aspects of surface modelling: (1) delineation of the (scale-dependent) geometrical boundary of a body via molecular considerations, (2) identification of the highly inhomogeneous interfacial region between a body and its exterior, and its modelling as a bidimensional continuum involving interfacial excess quantities, (3) the utility of co-ordinate-free notation for surfaces, and (4) the importance of surface effects for small-scale bodies exemplified within a thermoelastic context.      6. A New Rational Approach to Jet Noise Reduction      John M. Seiner 《Theoretical and Computational Fluid Dynamics》1998,10(1-4):373-383 This paper offers a new rational approach to jet noise suppression methodology. The tools required to implement such an approach are discussed. These involve the use of global velocimetry to measure those Lighthill turbulent stress tensor terms thought most important to noise generation. The framework of the dynamic systems model as it applies to the development of a noise suppression model is discussed, as well as the latest results on the development of an actuator for control input. The dynamic systems model offers a unique opportunity to study the influence of a concept noise reduction scheme on noise reduction through appropriate algorithms that relate turbulent jet processes to noise. Received 24 January 1997 and accepted 30 May 1997      7. An Algorithmic Approach to the Conley Index Theory      Marian Mrozek 《Journal of Dynamics and Differential Equations》1999,11(4):711-734 We introduce a class of representable sets which is closed under the operations of set theoretical union, intersection, difference, and topological interior and closure. We use this class to construct an algorithm which verifies if for a given dynamical system a given set is an isolating neighborhood. In the case of a positive answer the algorithm constructs an index pair.      8. A New Approach to Asymptotic Diagonalization of Linear Differential Systems      Sigrun I. Bodine  Robert J. Sacker 《Journal of Dynamics and Differential Equations》2000,12(1):229-245 We study the asymptotic diagonalization of a system consisting of an -matrix plus a finite number of -perturbations on an interval I 0=[t 0, ), where p, m i[1, ). Using linear skew-product flows and spectral theory, we show that if the unperturbed system has full spectrum over its omega-limit set, then the entire system is asymptotically diagonalizable almost everywhere.      9. A New Approach to Non-Isothermal Models for Nematic Liquid Crystals      Eduard?Feireisl  Michel?Frémond  Elisabetta?Rocca  Giulio?SchimpernaEmail author 《Archive for Rational Mechanics and Analysis》2012,205(2):651-672 We introduce a new class of non-isothermal models describing the evolution of nematic liquid crystals and prove their consistency with the fundamental laws of classical thermodynamics. The resulting system of equations captures all essential features of physically relevant models; in particular, the effect of stretching of the director field is taken into account. In addition, the associated initial-boundary value problem admits global-in-time weak solutions without any essential restrictions on the size of the initial data.      10. A New Representation Theorem for Elastic Constitutive Equations of Cubic Crystals      H. Xiao 《Journal of Elasticity》1998,53(1):37-45 The objective of this paper is to provide a new irreducible nonpolynomial representation for elastic constitutive equations of cubic crystals with the material symmetry group O or T d or O h . The presented result is expressed in terms of a generating set composed of nine polynomial tensor generators. It is simpler and more compact than a recent result in terms of a generating set composed of ten tensor generators. This revised version was published online in August 2006 with corrections to the Cover Date.      11. Complexity Versus Complex Systems: A New Approach to Scientific Discovery      Arecchi  F. T. 《Nonlinear dynamics, psychology, and life sciences》2001,5(1):21-35 Extraction of quantitative features from observations via suitable measuring devices M means that the words of science are coded as numbers, and the syntaxis is a set of mathematical rules. Once general premises are available all consequences can be worked in a purely deductive way. This characteristic of science displays two orders of drawbacks, namely, undecidability of deductive procedures, and intractability of computer modelings of complex situations. The way out of such a crisis consists in an adaptive strategy, that is, in a frequent readjustment of M suggested by the observed events. As a consequence, M provides different data streams (words) for the same observed events, as it is tuned to different resolutions. The adaptive strategy here introduced should by no means be confused with the adaptivity of a learning machine, which—inputted by a data stream—readjusts itself over a class of theoretical explanations in order to select the optimal one, thus providing knowledge conditional on the assigned input. On the contrary, physics aims at extracting regular patterns out of things, by a trial and error procedure which includes not only modifications of the explanations for fixed data sets, but also exploring different data sets via modified M's. This M-adjustment is a pre-linguistic endeavour, not expressible by a formal language. Such an essential characteristic of the physical program means that physics can not be performed by a machine.      12. An Integrated Approach to the Separation of Principal Surface Stresses Using Combined Thermo-Photo-Elasticity      R. J. Greene  E. A. Patterson 《Experimental Mechanics》2006,46(1):19-29 A new instrument capable of the full-field separation of principal stresses on the surface of a component is presented. The instrument combines the techniques of thermoelastic stress analysis and reflection photoelasticity in a single optical head, permitting the simultaneous capture of both data from the same point of view. A single strain witness coating is employed for the acquisition of both the thermoelastic and photoelastic data, which is both birefringent under applied stress conditions and opaque at the infrared wavelengths to which the thermoelastic analysis system is sensitive. This enables the combined technique to be performed continuously from the same surface during loading. The performance of the new instrument is validated in the analysis of a classical laboratory specimen of known geometry. Separated stress data from the experiment is compared to simulated data, demonstrating that the accuracy of the stress separation technique is comparable to that of the individual thermoelastic and photoelastic techniques, and it is concluded that combined thermo-photo-elasticity is a powerful tool for the experimental separation of principal surface stresses.      13. Modelling in Applied Sciences: A Kinetic Theory Approach      《Meccanica》2004,39(2):187-188       14. A New Approach to Characterizing the Equilibrium States of the Reynolds Stress Anisotropy in Homogeneous Turbulence      T. Jongen  T.B. Gatski 《Theoretical and Computational Fluid Dynamics》1998,11(1):31-47       15. Cauchy's Flux Theorem in Light of Geometric Integration Theory      G. Rodnay  R. Segev 《Journal of Elasticity》2003,71(1-3):183-203 This work presents a formulation of Cauchy's flux theory of continuum mechanics in the framework of geometric integration theory as formulated by H. Whitney and extended recently by J. Harrison. Starting with convex polygons, one constructs a formal vector space of polyhedral chains. A Banach space of chains is obtained by a completion process of this vector space with respect to a norm. Then, integration operators, cochains, are defined as elements of the dual space to the space of chains. Thus, the approach links the analytical properties of cochains with the corresponding properties of the domains in an optimal way. The basic representation theorem shows that cochains may be represented by forms. The form representing a cochain is a geometric analog of a flux field in continuum mechanics.      16. 边界元法边界层处理的一种新方法—拟多连域法   总被引：3，自引：0，他引：3   王元丰  邹永超 《应用力学学报》1997,14(1):124-127 提出了处理边界元法边界层问题的新方法。这种方法在求得边界未知量后，在距边界较远的域中构造一工作边界，将实际边界与工作边界看作一多连域问题求得工作边界的未知量。再将工作边界与边界层边界视为一多连域问题求解，可得到满足精度要求的边界附近点的位移与应力。这种方法理论简洁、计算方便、有效、精度高，对于需求边界上多点的未知量问题很具优越性      17. Correction to: A Novel Approach for the Determination of Surface Tilt Angles in Two-Dimensional Digital Image Correlation Experiments      Hijazi  A. L. 《Experimental Mechanics》2020,60(3):425-425 Experimental Mechanics - The original article has been corrected on the line “These two parameters are the average…” and on Eqs. 19, 20, 21 and 23.      18. Darcian Dynamics: A New Approach to the Mobilization of a Trapped Phase in Porous Media      P. Amili  Y. C. Yortsos 《Transport in Porous Media》2006,64(1):25-49 We develop a new approach, which we term Darcian Dynamics, to simulate two-phase (liquid-gas) flow in porous media, when the gas phase is disconnected in the form of ganglia. The method is based on the assumption of homogeneous fluid flow for the liquid, although it does allow for heterogeneous capillary thresholds due to the pore microstructure. Using techniques from potential theory, the hydrodynamic interaction between liquid and gas is expressed through an integral representation over the ganglia interfaces. We use a numerical method to solve the resulting integral equation, and explore conditions for the onset of ganglia mobilization as well as for subsequent events, such as break-up, coalescence and stranding. The interaction between the ganglia and the flowing phase is influenced by the capillary and gravity (Bond) numbers, and by geometric factors, such as size, orientation, and ganglia density. The latter effect depends on the hydrodynamic interaction in addition to the intuitively expected crowding effect.      19. 无源北斗导航定位系统初始校频的一种新方法      梁勇  柯熙政  邓方林 《中国惯性技术学报》2003,11(6):59-63 根据小波分析理论，针对无源北斗导航定位系统初始校频问题，提出了一种基于小波分析的信号特征提取算法。仿真结果表明该方法比较简单而切实可行，能够有效地对信号进行趋势估计，从而精确算得频标准确度。      20. A New and Efficient Approach to Determining the Rate of the Second-Order Symmetric Tensor and Its Isotropic Function      《Acta Mechanica Solida Sinica》2022,35(5):867-879

Some Fundamental Aspects of Surface Modelling

Attention is drawn to four aspects of surface modelling: (1) delineation of the (scale-dependent) geometrical boundary of a body via molecular considerations, (2) identification of the highly inhomogeneous interfacial region between a body and its exterior, and its modelling as a bidimensional continuum involving interfacial excess quantities, (3) the utility of co-ordinate-free notation for surfaces, and (4) the importance of surface effects for small-scale bodies exemplified within a thermoelastic context.     

A New Rational Approach to Jet Noise Reduction

This paper offers a new rational approach to jet noise suppression methodology. The tools required to implement such an approach are discussed. These involve the use of global velocimetry to measure those Lighthill turbulent stress tensor terms thought most important to noise generation. The framework of the dynamic systems model as it applies to the development of a noise suppression model is discussed, as well as the latest results on the development of an actuator for control input. The dynamic systems model offers a unique opportunity to study the influence of a concept noise reduction scheme on noise reduction through appropriate algorithms that relate turbulent jet processes to noise. Received 24 January 1997 and accepted 30 May 1997     

An Algorithmic Approach to the Conley Index Theory

We introduce a class of representable sets which is closed under the operations of set theoretical union, intersection, difference, and topological interior and closure. We use this class to construct an algorithm which verifies if for a given dynamical system a given set is an isolating neighborhood. In the case of a positive answer the algorithm constructs an index pair.     

A New Approach to Asymptotic Diagonalization of Linear Differential Systems

We study the asymptotic diagonalization of a system consisting of an -matrix plus a finite number of -perturbations on an interval I ₀=[t ₀, ), where p, m _i[1, ). Using linear skew-product flows and spectral theory, we show that if the unperturbed system has full spectrum over its omega-limit set, then the entire system is asymptotically diagonalizable almost everywhere.     

A New Approach to Non-Isothermal Models for Nematic Liquid Crystals

We introduce a new class of non-isothermal models describing the evolution of nematic liquid crystals and prove their consistency with the fundamental laws of classical thermodynamics. The resulting system of equations captures all essential features of physically relevant models; in particular, the effect of stretching of the director field is taken into account. In addition, the associated initial-boundary value problem admits global-in-time weak solutions without any essential restrictions on the size of the initial data.     

A New Representation Theorem for Elastic Constitutive Equations of Cubic Crystals

The objective of this paper is to provide a new irreducible nonpolynomial representation for elastic constitutive equations of cubic crystals with the material symmetry group O or T _d or O _h . The presented result is expressed in terms of a generating set composed of nine polynomial tensor generators. It is simpler and more compact than a recent result in terms of a generating set composed of ten tensor generators. This revised version was published online in August 2006 with corrections to the Cover Date.     

Complexity Versus Complex Systems: A New Approach to Scientific Discovery

Extraction of quantitative features from observations via suitable measuring devices M means that the words of science are coded as numbers, and the syntaxis is a set of mathematical rules. Once general premises are available all consequences can be worked in a purely deductive way. This characteristic of science displays two orders of drawbacks, namely, undecidability of deductive procedures, and intractability of computer modelings of complex situations. The way out of such a crisis consists in an adaptive strategy, that is, in a frequent readjustment of M suggested by the observed events. As a consequence, M provides different data streams (words) for the same observed events, as it is tuned to different resolutions. The adaptive strategy here introduced should by no means be confused with the adaptivity of a learning machine, which—inputted by a data stream—readjusts itself over a class of theoretical explanations in order to select the optimal one, thus providing knowledge conditional on the assigned input. On the contrary, physics aims at extracting regular patterns out of things, by a trial and error procedure which includes not only modifications of the explanations for fixed data sets, but also exploring different data sets via modified M's. This M-adjustment is a pre-linguistic endeavour, not expressible by a formal language. Such an essential characteristic of the physical program means that physics can not be performed by a machine.     

An Integrated Approach to the Separation of Principal Surface Stresses Using Combined Thermo-Photo-Elasticity

A new instrument capable of the full-field separation of principal stresses on the surface of a component is presented. The instrument combines the techniques of thermoelastic stress analysis and reflection photoelasticity in a single optical head, permitting the simultaneous capture of both data from the same point of view. A single strain witness coating is employed for the acquisition of both the thermoelastic and photoelastic data, which is both birefringent under applied stress conditions and opaque at the infrared wavelengths to which the thermoelastic analysis system is sensitive. This enables the combined technique to be performed continuously from the same surface during loading. The performance of the new instrument is validated in the analysis of a classical laboratory specimen of known geometry. Separated stress data from the experiment is compared to simulated data, demonstrating that the accuracy of the stress separation technique is comparable to that of the individual thermoelastic and photoelastic techniques, and it is concluded that combined thermo-photo-elasticity is a powerful tool for the experimental separation of principal surface stresses.     

Cauchy's Flux Theorem in Light of Geometric Integration Theory

This work presents a formulation of Cauchy's flux theory of continuum mechanics in the framework of geometric integration theory as formulated by H. Whitney and extended recently by J. Harrison. Starting with convex polygons, one constructs a formal vector space of polyhedral chains. A Banach space of chains is obtained by a completion process of this vector space with respect to a norm. Then, integration operators, cochains, are defined as elements of the dual space to the space of chains. Thus, the approach links the analytical properties of cochains with the corresponding properties of the domains in an optimal way. The basic representation theorem shows that cochains may be represented by forms. The form representing a cochain is a geometric analog of a flux field in continuum mechanics.     

边界元法边界层处理的一种新方法—拟多连域法

提出了处理边界元法边界层问题的新方法。这种方法在求得边界未知量后，在距边界较远的域中构造一工作边界，将实际边界与工作边界看作一多连域问题求得工作边界的未知量。再将工作边界与边界层边界视为一多连域问题求解，可得到满足精度要求的边界附近点的位移与应力。这种方法理论简洁、计算方便、有效、精度高，对于需求边界上多点的未知量问题很具优越性     

Correction to: A Novel Approach for the Determination of Surface Tilt Angles in Two-Dimensional Digital Image Correlation Experiments

Experimental Mechanics - The original article has been corrected on the line “These two parameters are the average…” and on Eqs. 19, 20, 21 and 23.     

Darcian Dynamics: A New Approach to the Mobilization of a Trapped Phase in Porous Media

We develop a new approach, which we term Darcian Dynamics, to simulate two-phase (liquid-gas) flow in porous media, when the gas phase is disconnected in the form of ganglia. The method is based on the assumption of homogeneous fluid flow for the liquid, although it does allow for heterogeneous capillary thresholds due to the pore microstructure. Using techniques from potential theory, the hydrodynamic interaction between liquid and gas is expressed through an integral representation over the ganglia interfaces. We use a numerical method to solve the resulting integral equation, and explore conditions for the onset of ganglia mobilization as well as for subsequent events, such as break-up, coalescence and stranding. The interaction between the ganglia and the flowing phase is influenced by the capillary and gravity (Bond) numbers, and by geometric factors, such as size, orientation, and ganglia density. The latter effect depends on the hydrodynamic interaction in addition to the intuitively expected crowding effect.     

无源北斗导航定位系统初始校频的一种新方法

根据小波分析理论，针对无源北斗导航定位系统初始校频问题，提出了一种基于小波分析的信号特征提取算法。仿真结果表明该方法比较简单而切实可行，能够有效地对信号进行趋势估计，从而精确算得频标准确度。