Infinitesimal Element Research Articles

A General Mathematical Method for Investigating the Thymic Microenvironment, Thymocyte Development, and Immunopathogenesis

T-lymphocyte (T-cell) development constitutes one of the basic and most vital processes in immunology. The process is profoundly affected by the thymic microenvironment, the dysregulation of which may be the pathogenesis or the etiology of some diseases. On the basis of a general conceptual framework, we have designed the first biophysical model to describe thymocyte development. The microclimate within the thymus, which is shaped by various cytokines, is first conceptualized into a growth field lambda and a differentiation field mu, under the influence of which the thymocytes mature. A partial differential equation is then derived through the analysis of an infinitesimal element of the flow of thymocytes. A general method is presented to estimate the two fields based on experimental data obtained by flow cytometric analysis of the thymus. Numerical examples are given for both normal and pathologic conditions. Our results are quite good, and even the time varying fields can be accurately estimated. Our method has demonstrated its great potential for the study of immunopathogenesis. The plan for implementation of the method is addressed.

Mechanics of Solids and Shells: Theories and Approximations

Mechanics of Solids and Shells: Theories and Approximations

A global and a local criterion in defining the tidal radius

AbstractThe basic theory of potential‐energy tensors related to heterogeneous spheres is reviewed, and the special case of truncated, singular, isothermal spheres is examined in detail. Special effort is devoted to a system made of two isothermal spheres, one completely lying within the other, the mass and the radius of the embedded sphere being negligible with respect to the mass and radius of the embedding sphere, and the radius of the embedded sphere being negligible with respect to the distance between the centres. The potential‐energy tensors related to the potential induced by the embedding sphere on the mass distribution of the embedded sphere, are expressed as the sum of two contributions: one, coming from the embedded sphere after collapse towards its centre, and one other, related to the actual mass distribution of the embedded sphere. Using the latter, both a global and a local criterion in defining the tidal radius of the embedded sphere, are formulated in connection with either the binding‐energy tensor or the virial‐energy tensor. In doing this, the tensor components along the axis joining the centre of the embedding and the embedded sphere, are considered. The global criterion is related to the whole, embedded sphere, while the local criterion is related to an infinitesimal mass element placed at the boundary of the embedded sphere, where the distance from the centre of the embedding sphere attains a maximum. The virial theorem in tensor form is splitted into two distinct expressions, related to orbital and intrinsic motions of the embedded sphere. Alternative criterions in defining the tidal radius of the embedded sphere, are formulated taking into consideration the centrifugal tensor potential and the tensor potential induced by orbital motions. With regard to a selected criterion, the tidal radius calculated with and without the inclusion of the centrifugal potential, exhibits a maximum variation by a factor of about two, related to circular orbits. An application is made, where the embedding and the embedded sphere are taken as representative of the Galaxy and a globular cluster, respectively. It is found that a stability region exists for both the global and the local criterion and, in addition, the instability first occurs at the perigalacticon, as expected in connection with instantaneous tidal radius. A powerlaw dependence of tidal radius from cluster mass and galactocentric distance, a*C ∝ M1/3CR2/30, is shown to be consistent with data from a sample of 16 objects investigated by Brosche et al. (1999). No significant correlation is found between the ratio of cluster radius to tidal radius and the orbital ratio of apogalacticon to perigalacticon, similar to averaged tidal radii defined by Brosche et al. (1999). An additional object, Pal 5, which is experiencing progressive disruption via tidal shocks during disk passages, is shown to be among the less bound (or more unbound) clusters, within the framework of the model. If the representation of globular clusters as isothermal spheres introduces only systematic errors in the ratio of cluster radius to tidal radius, aC/a*C = γ/η, then at least one other cluster, NGC5466 (which has the highest value between the sample objects), is inferred to undergo tidal disruption, in the model interpretation.

Viscous flow and deformation of regional metamorphic belts at convergent plate boundaries

Viscous flow and deformation of a Newtonian fluid between rigid boundaries have been modeled for investigating deformation of a forearc wedge and a regional metamorphic belt as a weak deforming zone between the plates at convergent boundaries. Three types of analytic formulations together with numerical formulations for a Newtonian fluid with a constant viscosity are used for investigating various flows and deformation: (1) flows induced by squeezing between two parallel boundaries, (2) flows induced by squeezing and heterogeneous mass influx in the wedge between two oblique boundaries (e.g., accretionary wedge over the subducting plate with the continental plate as a backstop), and (3) flows induced by dragging along the boundaries and by homogeneous mass influx in the wedge as in type 2. All cases include strike‐slip movements of the boundaries, which allows us to investigate three‐dimensional (3‐D) deformation (e.g., 3‐D corner flow). The corresponding flow, deformation of infinitesimal and finite elements, and the timescale of flow are calculated for each configuration with various mechanical boundary conditions. The model results show that configuration and mechanism of the flow can be inferred from the spatial variation of finite deformation (e.g., deformed radiolarians as a strain marker and geometry of folding as finite deformation of a large block). In particular, prolate strain nearly parallel to the strike of subduction zone, together with a large‐scale folding, can be a good indicator for deformation in the forearc wedge associated with 3‐D corner flow induced by oblique subduction. Deformation observed in the Cretaceous regional metamorphic belts in southwest Japan can be explained by this mechanism.

A coupled IEM/FEM approach for solving elastic problems with multiple cracks

In this paper, the detailed two-dimensional infinite element method (IEM) formulation with infinite element (IE)–finite element (FE) coupling scheme for investigating mode I stress intensity factor in elastic problems with imbedded geometric singularities (e.g. cracks) is presented. The IE–FE coupling algorithm is also successfully extended to solve multiple crack problems. In this method, the domain of the primary problem is subdivided into two sub-domains modeled separately using the IEM for the multiple crack sub-domain, and the FEM for the uncracked sub-domain. In the IE sub-domain, the similarity partition concept together with the IEM formulation are employed to automatically generate a large number of infinitesimal elements, layer by layer, around the tip of each crack. All degrees of freedom related to the IE sub-domain, except for those associated with the coupling interface, are condensed and transformed to form a finite master IE for each crack with master nodes on sub-domain boundary only. All of the stiffness matrices constructed in the IE sub-domains are assembled into the system stiffness matrix for the FE sub-domain. The resultant FE solution with a symmetrical stiffness matrix, having the singularity effect of imbedded cracks in IEs, is required only for solving multiple crack problems. Using these efficient numerical techniques a very fine mesh pattern can be established around each crack tip without increasing the degree of freedom of the global FEM solution. One is easily allowed to conduct parametric analyses for various crack sizes without changing the FE mesh. Numerical examples are presented to show the performance of the proposed method and compared with the corresponding known results where available.

Demonstration of balance-based space averaging and modelling in view of blading—diffuser interaction

Advanced experimental and numerical methods in the field of fluid dynamics and turbomachinery are increasingly successful in describing real flowfields, i.e. fields that are generally three-dimensional and unsteady. For many purposes, e.g. flow characterization, it is necessary to reduce these flowfields step by step to three-, two- or one-dimensional large-scale unsteady flowfields. This procedure permits a lower-level simulation of the flowfields. However, many averaging approaches are arbitrary or succeed in balancing the flowfields in only a few physical aspects. The first author has already shown the steps of a balance-based procedure that avoids this limitation. Small-scale time averaging of (probabilistically) turbulent inhomogeneities by means of irreversible and reversible small-scale time averaging processes on a threefold infinitesimal control volume element has already been demonstrated. The present paper demonstrates the balance-based procedure of space averaging. It is carried out by averaging generally three-dimensional small-scale time-averaged (deterministic) inhomogeneities using irreversible and reversible space averaging processes on onefold infinitesimal and finite control surfaces. The procedure is, similarly to small-scale time averaging, based on conservative and independent non-conservative small-scale time-averaged integral balance equations. The general concept is to represent all the relevant fluxes through the control surface by appropriate average quantities or numbers. The full use of the vector equations for the linear and angular momentum is important. One of the consequences in space averaging is the introduction of a wrench (parallel linear and angular momentum vectors), which is generally used only in mechanics for the reduction of force systems in space. The flowfield inhomogeneity is described on all dimensional levels via the diffusion intensity of the irreversible averaging process, and, only for space averaging, via the distance vector and the parameter of the wrench. A numerical example on different dimensional levels is presented in detail. The procedure also illustrates the basis of a new and more complete two-and one-dimensional large-scale unsteady theory generally in fluid dynamics and especially in turbomachinery.

Enhancement of Methane-Air Mixing Using Shock and Expansion Waves

A unified model of gaseous fuel and air mixing is applied here to study the use of shock waves for enhancement of mixing between methane and air. The model uses fuel mass fraction within infinitesimal fluid elements and the total derivative of this fraction with respect to time to measure the degree and rate of mixing, respectively. The model is accurate only for low-pressure combustors since it is based on the ideal gas law. The model is also limited to gaseous fuels that contain single chemical specie, or those that behave like single specie. The model presented here can be applied to any combustor geometry or operational conditions. Results show that mixing can be completed within the narrow region of the shock wave and therefore in a negligibly short time, if pressure, temperature, and velocity distributions within this region are optimized. Furthermore, the combined effects of air preheat and shock waves can enhance both mixing mechanisms with air penetration into the fuel and with fuel dispersion into the surrounding air. These results provide important guidelines for the mixing in supersonic combustors that are required to provide high efficiency and high intensity, while maintaining low levels of pollutants emission.

Surreal dimensions

In this paper we use Conway's surreal numbers to define a refinement of the box-counting dimension of a subset of a metric space. The surreal dimension of such a subset is well-defined in many cases in which the box-counting dimension is not. Surreal dimensions refine box-counting dimensions due to the fact that the class of surreal numbers contains infinitesimal elements as well as every real number. We compute the surreal dimensions of generalized Cantor sets, and we state some open problems.

Exact static solutions to piezoelectric smart beams including peel stresses: I: Theoretical formulation

This paper presents exact static solutions to smart beams with perfectly bonded or partially debonded piezoelectric (PZT) actuators and sensors including peel stresses. When a PZT patch is bonded on the surface of a beam, both shear and peel stresses exist in the adhesive between the PZT patch and the host beam. In Part I, non-dimensional differential governing equations for infinitesimal elements of the PZT patch and the host beam are formulated firstly, and then the ordinary differential equation (ODE) of the coupled shear and peel stresses are completely solved analytically. The exact solutions are applicable to smart beams with PZT actuators and sensors. When the PZT patches are used as actuators, the solutions give the formulations for the energy transferring shear and peel stresses in the adhesive actuated by the applied voltages, and for the actuated stress resultants and displacements in the host beam. When the PZT patches are used as sensors, the solutions give the shear and peel stresses in the adhesive caused by the loadings, and also the sensing electric charges. When PZT patches are not perfectly bonded to the host beam, i.e., partially debonded, the developed solutions can be tailored to the case of edge debondings, i.e., PZTs with the shortened length.

Non-linear static–dynamic finite element formulation for composite shells

Three non-linear finite element formulations for a composite shell are discussed. They are the simplified large rotation (SLR), the large displacement large rotation (LDLR), and the Jaumann analysis of general shells (JAGS). The SLR and the LDLR theories are based on total Lagrangian approach, and the JAGS is based on a co-rotational approach. Both the SLR and LDLR theories represent the in-plane strains exactly the same as Green's strain–displacement relations, whereas, only linear displacement terms are used to represent the transverse shear strain. However, a higher order kinematic through the thickness assumption is used in the SLR theory, which leads to parabolic transverse shear stress distribution compared to a first order kinematic through the thickness relationship used in the LDLR theory that leads to linear transverse shear stress distribution. Furthermore, the LDLR theory uses an Euler-like angle in the kinematics to account for the large displacement and rotation. The JAGS theory decomposes the deformation into stretches and rigid body rotations, where an orthogonal coordinate system translates and rotates with the deformed infinitesimal volume element. The Jaumann stresses and strains are used. Layer-wise stretching and shear warping through the thickness functions are used to model the three-dimensional behavior of the shell, where displacement and stress continuities are enforced along the ply interfaces. The kinematic behavior is related to the original undeformed coordinate system using the global displacements and their derivatives. Numerical analyses of composite shells are performed to compare the three theories. The commercial code ABAQUS is also used in this investigation as a comparison.

In General There Are No Unsheared Tetrads

Suppose the principal stretches are all different at a point P in a deformed body. In this case, it has been shown [1] that generally there is an infinity of non coplanar infinitesimal material line elements at P which remain unsheared following the deformation – that is, the angle between the arms of each pair of material line elements forming the triad remains unchanged. Here it is shown that in this case when all three principal stretches at P are different, there is no set of four infinitesimal material line elements, no three of which are coplanar, and such that the angle between each pair of the six pairs of material line elements is unchanged following the deformation. It is only when all three principal stretches at P are equal to each other, that there are unsheared tetrads at P, and in that case all tetrads are unsheared.

A Note on Maximum Shear

The problem of the determination at any point P in a body of that pair of infinitesimal material line elements which suffers the maximum shear in a deformation has been solved [1]. Here that problem is revisited and a short proof, of geometrical type, of the result is presented.

Matrix approach to Lagrangian fluid dynamics

A new approach to ideal-fluid hydrodynamics based on the notion of continuous deformation of infinitesimal material elements is proposed. The matrix approach adheres to the Lagrangian (material) view of fluid motion, but instead of Lagrangian particle trajectories, it treats the Jacobi matrix of their derivatives with respect to Lagrangian variables as the fundamental quantity completely describing fluid motion.A closed set of governing matrix equations equivalent to conventional Lagrangian equations is formulated in terms of this Jacobi matrix. The equation of motion is transformed into a nonlinear matrix differential equation in time only, where derivatives with respect to the Lagrangian variables do not appear. The continuity equation that requires constancy of the Jacobi determinant in time takes the form of an algebraic constraint on the Jacobi matrix. An accompanying linear consistency condition, which is responsible for the dependence on spatial variables and does not include time derivatives, ensures completeness of the system and reconstruction of the particle trajectories by the Jacobi matrix.A class of exact solutions to the matrix equations that describes rotational non-stationary three-dimensional motions having no analogues in the conventional formulations is also found and investigated. A distinctive feature of these motions is precession of vortex lines (rectilinear or curvilinear) around a fixed axis in space. Boundary problems for the derived exact solutions including matching of rotational and potential motions across the boundary of a vortex tube are addressed. In particular, for the cylindrical vortex of elliptical cross-section involved in three-dimensional precession, the outer potential flow is constructed and shown to be a non-stationary periodic straining flow at a large distance from the vortex axis.

Some remarks on Schanuel's conjecture

Schanuel's Conjecture is the statement: if x 1,…,x n ∈ C are linearly independent over Q , then the transcendence degree of Q(x 1,…,x n, exp(x 1),…, exp(x n)) over Q is at least n. Here we prove that this is true if instead we take infinitesimal elements from any ultrapower of C , and in fact from any nonarchimedean model of the theory of the expansion of the field of real numbers by restricted analytic functions.

A spectral element shallow water model on spherical geodesic grids

SUMMARY The spectral element method for the two-dimensional shallow water equations on the sphere is presented. The equations are written in conservation form and the domains are discretized using quadrilateral elements obtained from the generalized icosahedral grid introduced previously (Giraldo FX. Lagrange‐ Galerkin methods on spherical geodesic grids: the shallow water equations. Journal of Computational Physics 2000; 160: 336‐368). The equations are written in Cartesian co-ordinates that introduce an additional momentum equation, but the pole singularities disappear. This paper represents a departure from previously published work on solving the shallow water equations on the sphere in that the equations are all written, discretized, and solved in three-dimensional Cartesian space. Because the equations are written in a three-dimensional Cartesian co-ordinate system, the algorithm simplifies into the integration of surface elements on the sphere from the fully three-dimensional equations. A mapping (Song Ch, Wolf JP. The scaled boundary finite element method—alias consistent infinitesimal finite element cell method—for diffusion. International Journal for Numerical Methods in Engineering 1999; 45: 1403‐1431) which simplifies these computations is described and is shown to contain the Eulerian version of the method introduced previously by Giraldo (Journal of Computational Physics 2000; 160: 336‐368) for the special case of triangular elements. The significance of this mapping is that although the equations are written in Cartesian co-ordinates, the mapping takes into account the curvature of the high-order spectral elements, thereby allowing the elements to lie entirely on the surface of the sphere. In addition, using this mapping simplifies all of the three-dimensional spectral-type finite element surface integrals because any of the typical two-dimensional planar finite element or spectral element basis functions found in any textbook (for example, Huebner et al. The Finite Element Method for Engineers. Wiley, New York, 1995; Karniadakis GE, Sherwin SJ. Spectral:hp Element Methods for CFD. Oxford University ( ˘

Determination of Propane and Air Maximum Mixing Times

The model of gaseous fuel and air mixing, developed by the authors, is applied here to calculate maximum mixing times of propane and air. The degree of mixing is determined using the mass fraction of fuel while the rate of mixing is determined from the rate of this mass fraction. The values of both these parameters are local, i.e., measured within an infinitesimal element of fluid. A Eulerian representation is used. The model is based on the assumption that both fuel and air behave as a single chemical species. It is further assumed that pressure is low and only fuel and air are present within the fluid element. Under nonreacting conditions, the model is valid anywhere in the combustor. Under reacting conditions, the model is valid within those combustor regions where the fuel–air mixture is not flammable. The results of this analysis show that mixing times of propane and air are most reduced by high gradients of temperature and velocity, as long as these gradients provide in phase contribution. To a lesser degree, high gradients of pressure also help reduce mixing times. High initial pressure and temperature increase mixing time. Mixing with air penetration into the fuel flow is slower than with propane dispersion into the surrounding air. In general, the exact mixing time has to be determined numerically. Nevertheless, the analytical solutions included here provide maximum mixing times of propane and air under most conditions. These results provide important guidelines for the development of high intensity, high efficiency, and low emission combustors.

Nonstandard decision methods for the solvability of real polynomial equations

For a multivariate polynomial equation with coefficients in a computable ordered field, two criteria of this equation having real solutions are given. Based on the criteria, decision methods for the existence of real zeros and the semidefiniteness of binary polynomials are provided. With the aid of computers, these methods are used to solve several examples. The technique is to extend the original field involved in the question to a computable non-Archimedean ordered field containing infinitesimal elements.

Extension to linear dynamics for hybrid stress finite element formulation based on additional displacements

Based upon legitimate variational principles, one microscopic-macroscopic finite element formulation for linear dynamics is presented by Hybrid Stress Finite Element Method. The microscopic application of Geometric Perturbation introduced by Pian and the introduction of infinitesimal limit core element (Baby Element) have been consistently combined according to the flexible and inherent interpretation of the legitimate variational principles initially originated by Pian and Tong. The conceptual development based upon Hybrid Finite Element Method is extended to linear dynamics with the introduction of physically meaningful higher modes.

The scaled boundary finite element method?alias consistent infinitesimal finite element cell method?for diffusion

The scaled boundary finite element method, alias the consistent infinitesimal finite element cell method, is developed starting from the diffusion equation. Only the boundary of the medium is discretized with surface finite elements yielding a reduction of the spatial dimension by one. No fundamental solution is necessary, and thus no singular integrals need to be evaluated. Essential and natural boundary conditions on surfaces and conditions on interfaces between different materials are enforced exactly without any discretization. The solution of the function in the radial direction is analytical. This method is thus exact in the radial direction and converges to the exact solution in the finite element sense in the circumferential directions. The semi-analytical solution inside the domain leads to an efficient procedure to calculate singularities accurately without discretization in the vicinity of the singular point. For a bounded medium symmetric steady-state stiffness and mass matrices with respect to the degrees of freedom on the boundary result without any additional assumption. Copyright © 1999 John Wiley & Sons, Ltd.

Triangular stochastic matrices generated by infinitesimal elements

We show that each element in the semigroup S n of all n × n non-singular upper (or lower) triangular stochastic matrices is generated by the infinitesimal elements of S n, which form a cone consisting of all n × n upper (or lower) triangular intensity matrices.