Algebraic Boundary Research Articles

Cohesive crack propagation modelling in wood structures using BEM and the Tangent Operator Technique

Mechanical collapse in a wide range of materials occurs due to the growth of internal discontinuities. Such discontinuities, commonly denominated cracks, are explained consistently according to fracture mechanics theory. When the fracture process zone in front of the crack tip is sufficiently large, nonlinear mechanical effects appear and cannot be neglected. To allow a robust and general mechanical representation of such nonlinear phenomena, numerical techniques are required. In this context, this study presents an efficient nonlinear solution technique coupled to algebraic Boundary Element Method (BEM) equations to model the crack propagation process in anisotropic quasi-brittle bodies, using wood as a particular case. This nonlinear technique, called the Tangent Operator (TO), incorporates the derivative set of constitutive nonlinear laws into the algebraic BEM equations. The proposed nonlinear formulation was applied in mechanical analyses involving multi-crack growth and crack propagation in anisotropic media. The numerical results obtained by BEM/TO were compared with experimental and numerical responses available in the literature. In addition to the accuracy observed, the TO demonstrated faster convergence when compared with the classical approach.

Sampling and Reconstruction of Shapes With Algebraic Boundaries

We present a sampling theory for a class of binary images with finite rate of innovation (FRI). Every image in our model is the restriction of $\mathds{1}_{\{p\leq0\}}$ to the image plane, where $\mathds{1}$ denotes the indicator function and $p$ is some real bivariate polynomial. This particularly means that the boundaries in the image form a subset of an algebraic curve with the implicit polynomial $p$. We show that the image parameters --i.e., the polynomial coefficients-- satisfy a set of linear annihilation equations with the coefficients being the image moments. The inherent sensitivity of the moments to noise makes the reconstruction process numerically unstable and narrows the choice of the sampling kernels to polynomial reproducing kernels. As a remedy to these problems, we replace conventional moments with more stable \emph{generalized moments} that are adjusted to the given sampling kernel. The benefits are threefold: (1) it relaxes the requirements on the sampling kernels, (2) produces annihilation equations that are robust at numerical precision, and (3) extends the results to images with unbounded boundaries. We further reduce the sensitivity of the reconstruction process to noise by taking into account the sign of the polynomial at certain points, and sequentially enforcing measurement consistency. We consider various numerical experiments to demonstrate the performance of our algorithm in reconstructing binary images, including low to moderate noise levels and a range of realistic sampling kernels.

Algebraic boundaries of convex semi-algebraic sets

We study the algebraic boundary of a convex semi-algebraic set via duality in convex and algebraic geometry. We generalise the correspondence of facets of a polytope with the vertices of the dual polytope to general semi-algebraic convex sets. In this case, exceptional families of extreme points might exist and we characterise them semi-algebraically. We also give a strategy for computing a complete list of exceptional families, given the algebraic boundary of the dual convex set.

Heat-trace asymptotics for edge Laplacians with algebraic boundary conditions

We consider the Hodge-Laplace operator on manifolds with incomplete edge singularities and an intricate elliptic boundary value theory. We single out the class of algebraic self-adjoint extensions for the Hodge Laplacian. Our microlocal heat kernel construction for algebraic boundary conditions is guided by the method of signaling solutions by Mooers, though crucial arguments in the conical case obviously do not carry over to the setting of edges. We establish the heat kernel asymptotics for the algebraic extensions of the Hodge operator on edges, and elaborate on the exotic phenomena in the heat trace asymptotics which appear in the case of a non-Friedrichs extension.

Mapping properties of the heat operator on edge manifolds

We consider the heat operator acting on differential forms on spaces with complete and incomplete edge metrics. In the latter case we study the heat operator of the Hodge Laplacian with algebraic boundary conditions at the edge singularity. We establish the mapping properties of the heat operator, recovering and extending the classical results from smooth manifolds and conical spaces. The estimates, together with strong continuity of the heat operator, yield short‐time existence of solutions to certain semilinear parabolic equations. Our discussion reviews and generalizes earlier work by Jeffres and Loya.

Algebraic Boundaries of $$\mathrm{SO}(2)$$ SO ( 2 ) -Orbitopes

Let $$X\subset \mathbb{A }^{2r}$$XźA2r be a real curve embedded into an even-dimensional affine space. We characterise when the $$r$$rth secant variety to $$X$$X is an irreducible component of the algebraic boundary of the convex hull of the real points $$X(\mathbb{R })$$X(R) of $$X$$X. This fact is then applied to $$4$$4-dimensional $$\mathrm{SO}(2)$$SO(2)-orbitopes and to the so called Barvinok---Novik orbitopes to study when they are basic closed semi-algebraic sets. In the case of $$4$$4-dimensional $$\mathrm{SO}(2)$$SO(2)-orbitopes, we find all irreducible components of their algebraic boundary.

Choquet and Shilov Boundaries, Peak Sets, and Peak Points for Real Banach Function Algebras

Let be a compact Hausdorff space and let be a topological involution on . In 1988, Kulkarni and Arundhathi studied Choquet and Shilov boundaries for real uniform function algebras on . Then in 2000, Kulkarni and Limaye studied the concept of boundaries and Choquet sets for uniformly closed real subspaces and subalgebras of or . In 1971, Dales obtained some properties of peak sets and p-sets for complex Banach function algebras on . Later in 1990, Arundhathi presented some results on peak sets for real uniform function algebras on . In this paper, while we present a brief account of the work of others, we extend some of their results, either to real subspaces of or to real Banach function algebras on .

ORBITOPES

An orbitope is the convex hull of an orbit of a compact group acting linearly on a vector space. These highly symmetric convex bodies lie at the crossroads of several fields, including convex geometry, algebraic geometry, and optimization. We present a self-contained theory of orbitopes, with particular emphasis on instances arising from the groups SO(n) and O(n); these include Schur–Horn orbitopes, tautological orbitopes, Carathéodory orbitopes, Veronese orbitopes, and Grassmann orbitopes. We study their face lattices, algebraic boundaries, and representations as spectrahedra or projected spectrahedra.

A dynamical mother body in a Hele-Shaw problem

It is well-known that in the framework of the internal Hele-Shaw problem the fluid cannot be fully removed through a single point sink because of a cusp formation occurring prior to the moving boundary reaching the sink unless the initial domain is a circle with a sink located in its center. We give a definition of a dynamical mother body and use it for developing an algorithm of complete removal of a fluid droplet having an algebraic boundary. To illustrate our theory we consider examples where fluid can be completely removed through the sinks distributed along arcs of curves and/or points.

Critical review of conservation equations for two-phase flow in the U.S. NRC TRACE code

The field equations for two-phase flow in the computer code TRAC/ RELAP Advanced Computational Engine or TRACE are examined to determine their validity, their capabilities and limitations in resolving nuclear reactor safety issues. TRACE was developed for the NRC to predict thermohydraulic phenomena in nuclear power plants during operational transients and postulated accidents. TRACE is based on the rigorously derived and well-established two-fluid field equations for 1-D and 3-D two-phase flow. It is shown that: (1) The two-fluid field equations for mass conservation as implemented in TRACE are wrong because local mass balances in TRACE are in conflict with mass conservation for the whole reactor system, as shown in Section 3.1. (2) Wrong equations of motion are used in TRACE in place of momentum balances, compromising at branch points the prediction of momentum transfer between, and the coupling of, loops in hydraulic networks by impedance (form loss and wall shear) and by inertia and thereby the simulation of reactor component interactions. (3) Most seriously, TRACE calculation of heat transfer from fuel elements is incorrect for single and two-phase flows, because Eq. (3-4) of the TRACE Manual is wrong (see Section 5.2). (4) Boundary conditions for momentum and energy balances in TRACE are restricted to flow regimes 1 1 For classification of two-phase flow regimes see Delhaye et al. (1981, p. 47). with single-phase wall contact because TRACE lacks constitutive relations for solid–fluid exchange of momentum and heat in prevailing flow regimes. Without a quantified assessment of consequences from (3) to (4), predictions of phasic fluid velocities, fuel temperatures and important safety parameters, e.g., peak clad temperature, are questionable. Moreover, TRACE cannot predict 3-D single- or two-phase flows because: (5) incorrectly averaged equations are used for 3-D predictions, (6) fluid shear is ignored but needed to predict counter-current flows with the two-fluid model, and (7) fictitious body forces and fictitious distributed mass and heat sources are used to replace contact forces at the wall, mass injection and wall heat fluxes in 3-D mass, momentum and energy equations for both phases. No modeling error estimates are given in the TRACE Manual. (8) Imposed perfect mixing in every control volume causes artificial damping and disables TRACE to track reliably propagation of thermal and kinematic disturbances, thereby adversely affecting the prediction of nuclear fuel temperature. (9) According to its manual, TRACE relies on numerical diffusion far greater than physical dissipation by turbulence, to solve TRACE's ill-posed field equations and to compensate for missing fluid shear. TRACE with scale-sensitive numerical diffusion is tuned to match data from small-scale experiments. TRACE is therefore not reliable for analyzing full-size power plants. Numerical diffusion could also reduce the ability of TRACE to predict the onset of instability for reactors or test facilities entering large power and flow oscillations. Agreements of TRACE code results with experimental data from test facilities with isolated full-size components or with data from reduced-size integral-effect tests may be achieved by adjusting numerous coefficients, some of them scale-dependent (i.e., by tuning with documented non-physical coefficients in momentum balances, by spurious time-averaging (through time step adjustment) of algebraic boundary conditions, by built-in, non-physical space grid tuning in momentum balances and in mixture level predictions). Such results may not be applied reliably to full-size systems with realistic interactions between reactor components. TRACE may not be applicable to predict the outcome of postulated accidents for which validations are not possible (e.g., Anticipated Transients without Scram). New experimental techniques are needed first for closure of the two-fluid model. A new approach to TRACE modeling is needed. Recommendations are given for improved code documentation.

以代数曲线为边界的二维形体的Voronoi图

以代数曲线为边界的二维形体的Voronoi图

Sums of hermitian squares on pseudoconvex boundaries

We give an abstract characterization of all real algebraic subvarieties of complex ane space on which every positive polynomial is a sum of hermitian squares, and we find obstructions to this phenomenon. As a consequence we construct a strictly pseudoconvex domain with smooth algebraic boundary on which there exists a degree two positive polynomial which is not a sum of hermitian squares, answering thus in the negative a question of John D'Angelo.

Singular perturbation of a second-order three-point boundary value problem for nonlinear systems

This paper deals with the existence of solutions to a singularly perturbed second-order three-point boundary value problem for nonlinear differential systems. The authors construct an appropriate generalized lower- and upper-solution pair, a concept defined in this paper, and employ the Nagumo conditions and algebraic boundary layer functions to ensure the existence of solutions of the problem. The uniformly valid asymptotic estimate of the solutions is given as well. The differential systems have nonlinear dependence on all order derivatives of the unknown.

Optimal Control of Load Changes for Molten Carbonate Fuel Cell Systems: A Challenge in PDE Constrained Optimization

Molten carbonate fuel cells provide a promising technology for the operation of future stationary power plants. In order to enhance service life, a detailed understanding of the dynamical behavior of such fuel cell systems is necessary. In particular, fast load changes shall be simulated, (resp., optimized) without risking material stress due to the extreme temperature differences usually accompanying fast load changes. Fast load changes are important for daily operations in order to react on varying demands. Material stress may lead to irreparable damage of the fuel cell stack. For these contradicting goals, a family of hierarchically ordered mathematical models has been developed with the aim of simulating and optimizing the temporal and spatial dynamical behavior of the gas streams, chemical reactions, and potential fields within the fuel cells. Altogether, the most complicated system, which is investigated in the present paper, results in a Pareto-optimal control problem with constraints in form of a huge system of 28 partial differential algebraic equations and ordinary integro-differential algebraic equations and boundary conditions which are themselves partly given by an ordinary differential algebraic system of dimension 9. The PDEs are of parabolic and hyperbolic type; some are degenerate. Moreover, the variables involved in the different submodels of this fully coupled multiphysical system live on considerably different time scales. Optimal control results are presented for a compromise between sufficiently fast load changes and sufficiently small temperature differences within the cell's solid part by means of a specially tailored formulation of a chain of optimal control problems. This procedure benefits from the different time scales of the state variables and keeps the problem manageable and computable despite its tremendous complexity and scale, although standard numerical methods are employed.

Link concordance and generalized doubling operators

We introduce a technique for showing classical knots and links are not slice. As one application we show that the iterated Bing doubles of many algebraically slice knots are not topologically slice. Some of the proofs do not use the existence of the Cheeger-Gromov bound, a deep analytical tool used by Cochran-Teichner. We define generalized doubling operators, of which Bing doubling is an instance, and prove our nontriviality results in this more general context. Our main examples are boundary links that cannot be detected in the algebraic boundary link concordance group.

Constraint-Preserving Boundary Conditions for the Linearized Baumgarte-Shapiro-Shibata-Nakamura Formulation

We derive two sets of explicit homogeneous algebraic constraint-preserving boundary conditions for the second-order in time reduction of the linearized Baumgarte-Shapiro-Shibata-Nakamura (BSSN) system. Our second-order reduction involves components of the linearized extrinsic curvature only. An initial-boundary value problem for the original linearized BSSN system is formulated and the existence of the solution is proved using the properties of the reduced system. A treatment is proposed for the full nonlinear BSSN system to construct constraint-preserving boundary conditions without invoking the second order in time reduction. Energy estimates on the principal part of the BSSN system (which is first order in temporal and second order in spatial derivatives) are obtained. Generalizations to the case of nonhomogeneous boundary data are proposed.

2010 断面形状変化を伴う曲がり開水路内の乱流解析(G05-5 流体工学(5)乱流,21世紀地球環境革命の機械工学:人・マイクロナノ・エネルギー・環境)

Numerical analysis has been performed for the developing turbulent flow in a open-channel bend with changing cross section profile by using algebraic Reynolds stress model and boundary fitted coordinate system. Two kinds of calculations have been carried out, i.e., one is the calculation for smooth bed and the other is that for roughened bed. These calculated results are compared with the experimental data which is measured by a acoustic Doppler velocity profiler in order to examine the validity of the proposed numerical method and turbulent model. As a result of this analysis, it has been found that the calculated results for roughened bed show better agreement with the experimental data of the secondary vectors and time-averaged velocity than the calculated results for smooth bed.

Boundaries for algebras of holomorphic functions on Banach spaces

We study the relations between boundaries for algebras of holomorphic functions on Banach spaces and complex convexity of their balls. In addition, we show that the Shilov boundary for algebras of holomorphic functions on an order continuous sequence space $X$ is the unit sphere $S_X$ if $X$ is locally c-convex. In particular, it is shown that the unit sphere of the Orlicz-Lorentz sequence space $\lambda_{\varphi, w}$ is the Shilov boundary for algebras of holomorphic functions on $\lambda_{\varphi, w}$ if $\varphi$ satisfies the $\delta_2$-condition.

Singularly perturbed third-order boundary value problem for nonlinear systems

This paper deals with the existence of solutions to a singularly perturbed boundary value problem for third-order nonlinear differential systems. The authors construct an appropriate generalized lower solution–upper solution pair, a concept that is defined in this paper, and employ the method of descent, Nagumo conditions, algebraic boundary layer functions and Volterra type integral operator to guarantee the existence of solutions of the problem. The uniformly valid asymptotic estimate of the solutions is given as well. The differential systems has nonlinear dependence on all over order derivatives of the unknown.

Virtually Haken surgeries on once-punctured torus bundles

We describe a class $\mathcal{C}$ of punctured torus bundles such that, for each $M \in \mathcal{C}$, all but finitely many Dehn fillings on $M$ are virtually Haken. We show that $\mathcal{C}$ contains infinitely many commensurability classes, and we give evidence that $\mathcal{C}$ includes representatives of ``most'' commensurability classes of punctured torus bundles. In particular, we define an integer-valued complexity function on monodromies $f$ (essentially the length of the LR-factorization of $f_*$ in $PSL_2(\mathbb{Z})$), and use a computer to show that if the monodromy of $M$ has complexity at most 5, then $M$ is finitely covered by an element of $\mathcal{C}$. If the monodromy has complexity at most 12, then, with at most 36 exceptions, $M$ is finitely covered by an element of $\mathcal{C}$. We also give a method for computing ``algebraic boundary slopes'' in certain finite covers of punctured torus bundles.