scholarly journals The Multivariate Theory of Connections

Mathematics ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 296 ◽  
Author(s):  
Daniele Mortari ◽  
Carl Leake
Keyword(s):  
Final Section ◽  
Arbitrary Order ◽  
Mathematical Proof ◽  
Two Dimensions ◽  
Main Task ◽  
Boundary Constraints ◽  
Constrained Problems ◽  
Multivariate Theory ◽  
Analytical Expressions ◽  
Unconstrained Problems

This paper extends the univariate Theory of Connections, introduced in (Mortari, 2017), to the multivariate case on rectangular domains with detailed attention to the bivariate case. In particular, it generalizes the bivariate Coons surface, introduced by (Coons, 1984), by providing analytical expressions, called constrained expressions, representing all possible surfaces with assigned boundary constraints in terms of functions and arbitrary-order derivatives. In two dimensions, these expressions, which contain a freely chosen function, g ( x , y ) , satisfy all constraints no matter what the g ( x , y ) is. The boundary constraints considered in this article are Dirichlet, Neumann, and any combinations of them. Although the focus of this article is on two-dimensional spaces, the final section introduces the Multivariate Theory of Connections, validated by mathematical proof. This represents the multivariate extension of the Theory of Connections subject to arbitrary-order derivative constraints in rectangular domains. The main task of this paper is to provide an analytical procedure to obtain constrained expressions in any space that can be used to transform constrained problems into unconstrained problems. This theory is proposed mainly to better solve PDE and stochastic differential equations.

Analytical expressions for gravity anomalies due to homogeneous polyhedral bodies and translations into magnetic anomalies

Geophysics ◽  
1979 ◽  
Vol 44 (4) ◽  
pp. 730-741 ◽  
Author(s):  
M. Okabe
Keyword(s):  
Gravitational Potential ◽  
Computing Time ◽  
Gravity Anomalies ◽  
Magnetic Anomalies ◽  
Two Dimensions ◽  
First Derivative ◽  
Second Derivatives ◽  
Analytical Expressions ◽  
Derivatives Of ◽  
The Magnetic Field

Complete analytical expressions for the first and second derivatives of the gravitational potential in arbitrary directions due to a homogeneous polyhedral body composed of polygonal facets are developed, by applying the divergence theorem definitively. Not only finite but also infinite rectangular prisms then are treated. The gravity anomalies due to a uniform polygon are similarly described in two dimensions. The magnetic potential due to a uniformly magnetized body is directly derived from the first derivative of the gravitational potential in a given direction. The rule for translating the second derivative of the gravitational potential into the magnetic field component is also described. The necessary procedures for practical computer programming are discussed in detail, in order to avoid singularities and to save computing time.

scholarly journals Combining Interval Branch and Bound and Stochastic Search

2014 ◽  
Vol 2014 ◽  
pp. 1-15
Author(s):  
Dhiranuch Bunnag
Keyword(s):  
Genetic Algorithm ◽  
Simulated Annealing ◽  
Branch And Bound ◽  
Lower Cost ◽  
Search Algorithms ◽  
Stochastic Search ◽  
Constrained Problems ◽  
Interval Branch And Bound ◽  
Unconstrained Problems

This paper presents global optimization algorithms that incorporate the idea of an interval branch and bound and the stochastic search algorithms. Two algorithms for unconstrained problems are proposed, the hybrid interval simulated annealing and the combined interval branch and bound and genetic algorithm. The numerical experiment shows better results compared to Hansen’s algorithm and simulated annealing in terms of the storage, speed, and number of function evaluations. The convergence proof is described. Moreover, the idea of both algorithms suggests a structure for an integrated interval branch and bound and genetic algorithm for constrained problems in which the algorithm is described and tested. The aim is to capture one of the solutions with higher accuracy and lower cost. The results show better quality of the solutions with less number of function evaluations compared with the traditional GA.

scholarly journals ROTATING FERMIONS IN TWO DIMENSIONS: A THOMAS–FERMI APPROACH

2001 ◽  
Vol 15 (19n20) ◽  
pp. 2799-2810
Author(s):  
SANKALPA GHOSH ◽  
M. V. N. MURTHY ◽  
SUBHASIS SINHA
Keyword(s):  
Ground State ◽  
Analytical Approach ◽  
State Energy ◽  
Critical Size ◽  
Energy Functional ◽  
Two Dimensions ◽  
Fermionic System ◽  
Thomas Fermi ◽  
Fermionic Systems ◽  
Analytical Expressions

Properties of confined mesoscopic systems have been extensively studied numerically over recent years. We discuss an analytical approach to the study of finite rotating fermionic systems in two dimension. We first construct the energy functional for a finite fermionic system within the Thomas–Fermi approximation in two dimensions. We show that for specific interactions the problem may be exactly solved. We derive analytical expressions for the density, the critical size as well as the ground state energy of such systems in a given angular momentum sector.

Syntactic Argumentation

2019 ◽  
pp. 20-735
Author(s):  
Bas Aarts
Keyword(s):  
Case Studies ◽  
Final Section ◽  
Two Dimensions ◽  
English Grammar

The aim of this chapter is to discuss the role of syntactic argumentation in the description of English grammar. The chapter first discusses some of the general principles that are used in syntactic argumentation, namely economy and elegance, which are regarded as two dimensions of simplicity. These principles are then exemplified in a number of case studies. The final section discusses how argumentation can be used to establish constituency in clauses.

The Theoretical Investigation on Critical Buckling Stress of Graphene Nanosheets

2016 ◽  
Vol 859 ◽  
pp. 79-84
Author(s):  
Li Jun Zhou ◽  
Jian Gao Guo ◽  
Bao Long Li
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Graphene Nanosheets ◽  
Structural Mechanics ◽  
Elastic Buckling ◽  
Boundary Constraints ◽  
Buckling Stress ◽  
Good Consistency ◽  
Analytical Expressions ◽  
Geometrical Dimensions

The elastic buckling behaviors of graphene nanosheets are investigated via molecular structural mechanics based finite element method. The size-and chirality-dependent critical buckling stresses of monolayer and bilayer graphene nanosheets are calculated for different geometrical dimensions and boundary constraints, respectively. By analogy with classical buckling theory of elastic plate, the analytical expressions of critical buckling stress are derived for the graphene nanosheets with different boundary constraints, and the comparisons of analytical results with the counterparts obtained by molecular structural mechanics simulation show a good consistency.

scholarly journals MATHEMATICAL MODELING OF COLD PRESSING THE SHEET COMPOSITE

2020 ◽  
pp. 55-64
Author(s):  
B. M. Kumitskiy ◽  
N. A. Savrasova ◽  
V. N. Melkumov ◽  
Ye. S. Aralov
Keyword(s):  
Mathematical Model ◽  
Two Dimensions ◽  
Plastic Layer ◽  
Plasticity Condition ◽  
Interior Decoration ◽  
Linear Distribution ◽  
Cold Pressing ◽  
Approach Speed ◽  
Analytical Expressions ◽  
Construction Works

Statement of the problem. The article examines the problem of cold pressing, which is the most important technological component in the production of sheet composite, which is widely studied in the repair and construction works in the interior decoration of residential and industrial premises. The solution to this problem is carried out on the basis of a physical and mathematical model under the assumption that the rheological properties of the deformable medium correspond to the principles of ideal plasticity and a flat deformable state. Within the framework of the problem, in two dimensions of quasistatic compression between absolutely rigid parallel-approaching plates of a thin ideally plastic layer, the stress-strain state of a composite medium is studied. It is believed that in the absence of volumetric loads, the condition of incompressibility of the medium and the associated flow law is fulfilled. Based on the hypothesis of the linear distribution of tangential stresses over the thickness of the deformable layer, analytical expressions for the statistical and kinematic characteristics of the deformation are obtained, and the condition at the edges of the rough plates makes it possible to determine the coefficient of slip thorns, which makes it possible to control the pressing process.Results and conclusions. It was established that the components of the strain rate are directly proportional to the plate approach speed, and the normal stresses acting in the pressing direction are independent of the loading speed, decreasing in magnitude from the center to the periphery.Keywords: yield strength, pressing, plasticity condition, mathematical model.

Stationary disk assemblies in a ternary system with long range interaction

2019 ◽  
Vol 21 (06) ◽  
pp. 1850046 ◽  
Author(s):  
Xiaofeng Ren ◽  
Chong Wang
Keyword(s):  
Ternary System ◽  
Long Range ◽  
Mathematical Proof ◽  
Two Dimensions ◽  
Domain Growth ◽  
Type I ◽  
Type Ii ◽  
Range Interaction ◽  
Stationary Disk ◽  
Long Range Interaction

The free energy of a ternary system, such as a triblock copolymer, is a sum of two parts: an interface energy determined by the size of the interfaces separating the micro-domains of the three constituents, and a long range interaction energy that serves to prevent unlimited micro-domain growth. In two dimensions a parameter range is identified where the system admits stable stationary disk assemblies. Such an assembly consists of perturbed disks made from either type-I constituent or type-II constituent. All the type-I disks have approximately the same radius and all the type-II disks also have approximately the same radius. The locations of the disks are determined by minimization of a function. Depending on the parameters, the disks of the two types can be mixed in an organized way, or mixed in a random way. They can also be fully separated. The first scenario offers a mathematical proof of the existence of a morphological phase for triblock copolymers conjectured by polymer scientists. The last scenario shows that the ternary system is capable of producing two levels of structure. The primary structure is at the microscopic level where disks form near-perfect lattices. The secondary structure is at the macroscopic level forming two large regions, one filled with type-I disks and the other filled with type-II disks. A macroscopic, circular interface separates the two regions.

scholarly journals Selected Applications of the Theory of Connections: A Technique for Analytical Constraint Embedding

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 537 ◽  
Author(s):  
Hunter Johnston ◽  
Carl Leake ◽  
Yalchin Efendiev ◽  
Daniele Mortari
Keyword(s):  
Differential Equations ◽  
Inequality Constraints ◽  
Variational Problems ◽  
Linear Constraint ◽  
Mathematical Framework ◽  
Constrained Problems ◽  
Collocation Points ◽  
Constraint Embedding ◽  
New Applications ◽  
Unconstrained Problems

In this paper, we consider several new applications of the recently introduced mathematical framework of the Theory of Connections (ToC). This framework transforms constrained problems into unconstrained problems by introducing constraint-free variables. Using this transformation, various ordinary differential equations (ODEs), partial differential equations (PDEs) and variational problems can be formulated where the constraints are always satisfied. The resulting equations can then be easily solved by introducing a global basis function set (e.g., Chebyshev, Legendre, etc.) and minimizing a residual at pre-defined collocation points. In this paper, we highlight the utility of ToC by introducing various problems that can be solved using this framework including: (1) analytical linear constraint optimization; (2) the brachistochrone problem; (3) over-constrained differential equations; (4) inequality constraints; and (5) triangular domains.

Seismic hazard induced by mechanically interactive fault segments

1993 ◽  
Vol 83 (2) ◽  
pp. 436-449
Author(s):  
C. Allin Cornell ◽  
Shen-Chyun Wu ◽  
Steven R. Winterstein ◽  
James H. Dieterich ◽  
Robert W. Simpson
Keyword(s):  
Final Section ◽  
Three Dimensional ◽  
Earthquake Risk ◽  
Physical Interaction ◽  
Earthquake Recurrence ◽  
Induced Stress ◽  
Stress Changes ◽  
Temporal And Spatial ◽  
Gross Behavior ◽  
Analytical Expressions

Abstract This paper presents a phenomenological stochastic model for earthquake recurrence processes involving physical interaction among fault segments. Slip on one segment may reduce (or increase) the time to the next event on another segment or possibly induce an immediate slip on that segment as well. The gross behavior of this model is first observed through simulations; temporal and spatial disorder are observed even when the stochastic aspects are minimized. To estimate the strength of these interactions, we derive factors from the output of three-dimensional elastic dislocation analyses, relating induced stress changes to temporal changes in next-event dates. In a final section, we derive approximate analytical expressions and numerical results for future probabilistic earthquake risk and site hazard, conditional on the elapsed times since events on all relevant fault segments and on the number of events since that may have caused stress changes (interactions).

Constrained Optimization by the α Constrained Particle Swarm Optimizer

2005 ◽  
Vol 9 (3) ◽  
pp. 282-289 ◽  
Author(s):  
Tetsuyuki Takahama ◽  
◽  
Setsuko Sakai ◽  
Keyword(s):  
Constrained Optimization ◽  
Optimization Problems ◽  
Particle Swarm ◽  
Satisfaction Level ◽  
Particle Swarm Optimizer ◽  
Constrained Optimization Problems ◽  
Swarm Optimization ◽  
Constrained Problems ◽  
Α Level ◽  
Unconstrained Problems

In this study, α constrained particle swarm optimizer αPSO, which is the combination of the α constrained method and particle swarm optimization, is proposed to solve constrained optimization problems. The α constrained methods can convert algorithms for unconstrained problems to algorithms for constrained problems using the α level comparison, which compares the search points based on the satisfaction level of constraints. In the αPSO, the agents who satisfy the constraints move to optimize the objective function and the agents who don't satisfy the constraints move to satisfy the constraints. The effectiveness of the αPSO is shown by comparing the αPSO with GENOCOP5.0, and other PSO-based methods on some nonlinear constrained problems.

Sign in / Sign up

Export Citation Format

Share Document