CONSTRUCTION OF A PRECONDITIONER FOR DOMAIN DECOMPOSITION METHODS WITH POLYNOMIAL LAGRANGIAN MULTIPLIERS

In this paper we consider domain decomposition methods with polynomial Lagrangian multipliers to two-dimentional elliptic problems, and construct a kind of simple preconditioners for the corresponding interface equation. It will be shown that condition number of the resulting preconditioned interface matrix is almost optimal (namelys it has only logarithmic growth with dimension of the local interface space).

Dark Energy from Kaluza-Klein Spacetime and Noether’s Theorem via Lagrangian Multiplier Method

The supposedly missing dark energy of the cosmos is found quantitatively in a direct analysis without involving ordinary energy. The analysis relies on five dimensional Kaluza-Klein spacetime and a Lagrangian constrained by an auxiliary condition. Employing the Lagrangian multiplier method, it is found that this multiplier is equal to the dark energy of the cosmos and is given by where E is energy, m is mass, c is the speed of light, and λ is the Lagrangian multiplier. The result is in full agreement with cosmic measurements which were awarded the 2011 Nobel Prize in Physics as well as with the interpretation that dark energy is the energy of the quantum wave while ordinary energy is the energy of the quantum particle. Consequently dark energy could not be found directly using our current measurement methods because measurement leads to wave collapse leaving only the quantum particle and its ordinary energy intact.

SUPER EFFICIENCY IN THE NEARLY CONE-SUBCONVEXLIKE VECTOR OPTIMIZATION WITH SET-VALUED FUNCTIONS

Some properties for convex cones are discussed, which are used to obtain an equivalent condition and another important property for nearly cone-subconvexlike set-valued functions. Under the nearly cone-subconvexlikeness, some characterizations of the super efficiency are given in terms of scalarization and Lagrangian multipliers. Related results are generalized.

BENSON PROPER EFFICIENCY IN THE NEARLY CONE-SUBCONVEXLIKE VECTOR OPTIMIZATION WITH SET-VALUED FUNCTIONS

Some properties of convex cones are obtained and are used to derive several equivalent conditions as well as another important property for nearly cone-subconvexlike set-valued functions. Under the assumption of nearly cone-subconvexlikeness,a Lagrangian multiplier theorem on Benson proper efficiency is presented. Related results are generalized.

On Chien's question to the Hu-Washizu three-field functional and variational principle

There is an open question,namely Chien’s question,in construction of a generalized functional in elasticity,i.e.,why the stress-strain relation can still be derived from the Hu-Washizu generalized variational principle while the Lagrangian multiplier method is applied in vain?This study shows that the generalized variational principle can only be understood and implemented correctly within the framework of thermodynamics.This investigation finds that as long as the functional has one of the combinations(A(ε_(ij))-σ_(ij)ε_(ij))or(B(σ_(ij))-σ_(ij)ε_(ij)),its corresponding variational principle will produce the stress-strain relation without the need to introduce extra constraints by the Lagrangian multiplier method.This research proves that the Hu-Washizu functionalΠ(u_(ij),ε_(ij),σ_(ij))is real three-field functional,and resolves the historic academic controversy on the issue of constructing a three-field functional.

STUDY ON THE PROBLEMS OF CONTACT IN THE NUMERICAL SIMULATION OF SHEET FORMING

This study is concerned with the problems of contact in the process of numerical simulation of sheet metal forming in rigid visco-plastic shell FEM. In respect of analysis of sheet deep drawing process,for the tool model described by triangular elements, a kind of contact judging algorithm about the correlation between the node of deformed mesh and the triangular element of a tool is presented. In SPF/DB Lagrangian multiplier method is adopted to solve the contact problem between deformed meshes, and a new reliable practical dynamic contact checking algorithm is presented. As computation examples, the simulation results of metal sheet deep drawing and SPF/DB are introduced in this paper.

Contribution of Disclination Lines to Free Energy of 2-Dimensional Liquid Crystals in Single-Elastic Constant Approximation

In light of the φ-mapping method, the contribution of disclination lines to the free energy density of 2-dimensional liquid crystals is studied in the single-elastic constant approximation. It is pointed out that, compared with the previous theory, the free energy density can be divided into two parts. One is the usual distorted energy density of director field around the disclination lines. The other is the free energy density of the disclination lines themselves which is centralized at the disclination lines and topoligically quantized in a unit of 1/2kπ. The topological quantum numbers are determined by the Hopf indices and Brouwer degrees of the director field at the disclination lines, i.e., the disclination strength. From the method of Lagrangian multipliers, the equilibrium equation and the molecular field of 2-dimensional liquid crystals are also obtained. It is shown that the physical meaning of the Lagrangian multiplier is just the distorted energy density. Key words director field - disclination line - free energy - Lagrangian multiplier PASC2001 64.70.Md - 02.40.-k

VARIATIONAL PRINCIPLES IN HYDRODYNAMICS OF A NON-NEWTONIAN FLUID

In this paper, the principle of maximum power losses for the incompressible viscous fluid proposed by professor Chien Weizang in reference [1] is further extended to the hydrodynamic problem of the non-Newtonian fluid with constitutive law expressed as epsilon(y) = partial derivative tau/partial derivative sigma'(y). The constraint conditions of variation are eliminated by the method of identified Lagrangian multiplier and a generalized variational principle is established.

Strict Efficiency in Vector Optimization with Ic-cone-convexlike Set-valued Functions

An important property of ic-cone-convexlike set-valued functions is obtained in this paper. Under the assumption of ic-cone-convexlikeness, the scalarization theorem and the Lagrange multiplier theorem for strict efficient solution are derived, respectively.

Numerical simulation of square shaped particle sedimentation

Direct numerical simulations of square particle sedimentation in a viscous incompressible fluid are presented to examine the effects of sharp edges for various particle-to-fluid density ratios and different initial inclination angles.The settling process exhibits four different motion regimes;non-oscillatory,non-uniform centered oscillatory,uniform off-centered oscillatory,and non-uniform off-centered(irregular)oscillatory.At moderate density ratios,we observe that the rotational motion of settling square particle varies with the initial inclination angle.We employ Dynamic Time Warping and Fast Fourier Transform to analyze translational and angular velocities.Oscillating systems are described using phase-space plots,and a chaotic phenomenon is observed at a higher density ratio.

Left-Invariant Minimal Unit Vector Fields on the Solvable Lie Group

Bozek(1980)has introduced a class of solvable Lie groups Gn with arbitrary odd dimension to construct irreducible generalized symmetric Riemannian space such that the identity component of its full isometry group is solvable.In this article,the authors provide the set of all left-invariant minimal unit vector fields on the solvable Lie group Gn,and give the relationships between the minimal unit vector fields and the geodesic vector fields,the strongly normal unit vectors respectively.

Investigating and Mitigating Failure Modes in Physics-Informed Neural Networks(PINNs)

This paper explores the difficulties in solving partial differential equations(PDEs)using physics-informed neural networks(PINNs).PINNs use physics as a regularization term in the objective function.However,a drawback of this approach is the requirement for manual hyperparameter tuning,making it impractical in the absence of validation data or prior knowledge of the solution.Our investigations of the loss landscapes and backpropagated gradients in the presence of physics reveal that existing methods produce non-convex loss landscapes that are hard to navigate.Our findings demonstrate that high-order PDEs contaminate backpropagated gradients and hinder convergence.To address these challenges,we introduce a novel method that bypasses the calculation of high-order derivative operators and mitigates the contamination of backpropagated gradients.Consequently,we reduce the dimension of the search space and make learning PDEs with non-smooth solutions feasible.Our method also provides a mechanism to focus on complex regions of the domain.Besides,we present a dual unconstrained formulation based on Lagrange multiplier method to enforce equality constraints on the model’s prediction,with adaptive and independent learning rates inspired by adaptive subgradient methods.We apply our approach to solve various linear and non-linear PDEs.

Henig Efficiency in Vector Optimization with Nearly Cone-subconvexlike Set-valued Functions

In this paper, we study Henig efficiency in vector optimization with nearly cone-subconvexlike set-valued function. The existence of Henig efficient point is proved and characterization of Henig efficiency is established using the method of Lagrangian multiplier. As an interesting application of the results in this paper, we establish a Lagrange multiplier theorem for super efficiency in vector optimization with nearly conesubconvexlike set-valued function.

Theoretical investigations on lattice Boltzmann method:an amended MBD and improved LBM

This paper presents theoretical investigations of lattice Boltzmann method(LBM)to develop a completed LBM theory.Based on H-theorem with Lagrangian multiplier method,an amended theoretical equilibrium distribution function(EDF)is derived,which modifies the current Maxwell–Boltzmann distribution(MBD)to include the total internal energy as its parameter.This modification allows the three conservation laws derived directly from lattice Boltzmann equation(LBE)without additional small-parameter expansions adopted in references.From this amended theoretical EDF,an improved LBM is developed,in which the total internal energy like the mass density and mean velocity is a new macroscopic variable to be updated for different times and cells during simulations.The developed method provides a means to consider external forces and energy generation sources as generalised forces in LBM simulations.The corresponding model and implementation process of the improved LBM are presented with its performance theoretically investigated.Analytically hand-workable examples are given to illustrate its applications and to confirm its validity.The paper will excite more researchers and scientists of this area to numerically practice the new theory and method dealing with complex physical problems,from which it is expected to further advance LBM benefiting science and engineering.