Minimum Variational Principle Research Articles

A remark on a strong minimum condition of a fractal variational principle

The fractal variational principle gives a good physical understanding of a discontinuous problem in an energy way, and it is a good tool to revealing the physical phenomenon which cannot be done by the traditional variational theory. A minimum variational principle is very important in ensuring the convergence of artificial intelligence algorithms for numerical simulation and image processing. The strong minimum condition of a fractal variational principle in a fractal space is discussed, and two examples are given to illustrate its simplicity and feasibility.

On some qualitative results in thermodynamics of Cosserat bodies

AbstractThis paper deals with the linear theory of thermoelastic Cosserat bodies. At the beginning, we formulate the mixed initial-boundary value problem in this context and obtain new theorems of reciprocity in the thermodynamics theory of these media. Then we prove that these new reciprocity relations imply the uniqueness of solution of the mixed problem. Based on the same reciprocal relations, we establish a minimum variational principle, which generalizes those from the theory of classical thermoelasticity.

On a strong minimum condition of a fractal variational principle

Variational theory is the theoretical basis for various numerical methods. A minimum variational principle plays an important role in guaranteeing the convergence of the numerical algorithm, and the Weierstrass theorem is used to judge the minimum. This paper suggests a fractal modification of Weierstrass function for fractal variational principles, and a strong minimum condition is given. The Hamilton least-action principle for the Duffing oscillator in a fractal space is used as an example to show the correctness of the suggested strong minimum condition.

Effective preconditioning for ab initio ground state energy minimization with non-orthogonal localized molecular orbitals

The non-orthogonal localized molecular orbital (NOLMO) is the most localized representation of electronic degrees of freedom. As such, NOLMOs are thus potentially the most efficient for linear-scaling calculations of electronic structures for large systems. However, direct ab initio calculations with NOLMO have not been fully implemented and widely used, partly because of the slow convergence issue in the optimization of NOLMO. Towards realizing the potential of NOLMO for large systems, we applied an energy minimum variational principle for carrying out ab initio self-consistent-field (SCF) calculations with NOLMOs. We developed an effective preconditioning approach using the diagonal part of the second order derivatives and show that the convergence of the energy optimization is significantly improved. The speed of convergence of the energy and density are comparable with that of the conventional SCF approach, thus paving the way for the optimization of NOLMO in linear scaling calculations for large systems.

A unified residual-based thermodynamic framework for strain gradient theories of plasticity

A unified thermodynamic framework for gradient plasticity theories in small deformations is provided, which is able to accommodate (almost) all existing strain gradient plasticity theories. The concept of energy residual (the long range power density transferred to the generic particle from the surrounding material and locally spent to sustain some extra plastic power) plays a crucial role. An energy balance principle for the extra plastic power leads to a representation formula of the energy residual in terms of a long range stress, typically of the third order, a macroscopic counterpart of the micro-forces acting on the GNDs (Geometrically Necessary Dislocations). The insulation condition (implying that no long range energy interactions are allowed between the body and the exterior environment) is used to derive the higher order boundary conditions, as well as to ascertain a principle of the plastic power redistribution in which the energy residual plays the role of redistributor and guarantees that the actual plastic dissipation satisfies the second thermodynamics principle. The (nonlocal) Clausius–Duhem inequality, into which the long range stress enters aside the Cauchy stress, is used to derive the thermodynamic restrictions on the constitutive equations, which include the state equations and the dissipation inequality. Consistent with the latter inequality, the evolution laws are formulated for rate-independent models. These are shown to exhibit multiple size effects, namely (energetic) size effects on the hardening rate, as well as combined (dissipative) size effects on both the yield strength (intrinsic resistance to the onset of plastic strain) and the flow strength (resistance exhibited during plastic flow). A friction analogy is proposed as an aid for a better understanding of these two kinds of strengthening effects. The relevant boundary-value rate problem is addressed, for which a solution uniqueness theorem and a minimum variational principle are provided. Comparisons with other existing gradient theories are presented. The dissipation redistribution mechanism is illustrated by means of a simple shear model.

Minimum variational principles for time-harmonic waves in a dissipative medium and associated variational principles of Hashin–Shtrikman type

Minimization variational principles for linear elastodynamic, acoustic or electromagnetic time-harmonic waves in dissipative media were obtained by Milton et al. (Milton et al. 2009 Proc. R. Soc. A 465 , 367–396 ( doi:10.1098/rspa.2008.0195 )), generalizing the quasistatic variational principles of Cherkaev and Gibiansky (Cherkaev & Gibiansky 1994 J. Math. Phys. 35 , 127–145 ( doi:10.1063/1.530782 )). Here, a further generalization is made to allow for a much wider variety of boundary conditions, and in particular Dirichlet and Neumann boundary conditions. In addition minimization or maximization principles of the Hashin–Shtrikman type, incorporating ‘polarization fields’, are developed. The corresponding principles for static problems have found substantial use in bounding the effective static properties of composite materials. The new dynamical principles offer the prospect of developing bounds on the effective dynamic properties of such materials.

Linear-scaling quantum calculations using non-orthogonal localized molecular orbitals

An absolute energy minimum variational principle is used for carrying out linear-scalingcalculations with non-orthogonal localized orbitals. Comparing with results based onorthogonal localized molecular orbitals, the method is shown to give significantly moreaccurate results when the localized molecular orbitals are allowed to be non-orthogonal.This is made possible by introducing a second minimization for approximatingthe inverse overlap matrix. We also show how an exact line search may be usedefficiently with the conjugate gradient method for minimizing the energy functional.

Variational principles for softening gradient dependent plasticity

Abstract A gradient dependent plasticity model is analysed in which the yield function not only depends on the stress tensor and an invariant plastic strain measure, but also on the spatial derivatives of the latter quantity. A minimum variational principle is obtained by analysing the structure discretized into sufficiently small plastic elements and thus the uniqueness of FE solutions can be ensured for softening plasticity. The positive definiteness of the variational functional is determined only by the sign of the diffuse coefficient associated with the highest order derivative of the plastic multiplier in the yield function. In order to adopt mix/hybrid elements, several general variational principles are obtained by relaxing the associated constraints on the displacement–strain–stress and/or plastic multiplier-gradient-radiation fields.

The Relationship Between Various Methods of Solution of the Lippmann-Schwinger Equation

The generalized Schwinger variational principle is shown to be equivalent to one of the three versions of the Rayski method. In the case of a spherically symmetric potential the Kohn-type variational principle is introduced and the relationship between methods mentioned above is found. Another version of the Rayski method is connected to a minimum variational principle and yields a convergent process for computing the exact solution of the Lippmann-Schwinger equation for the short-range potentials.

Creeping motion of a sphere through a Bingham plastic

A solid sphere falling through a Bingham plastic moves in a small envelope of fluid with shape that depends on the yield stress. A finite-element/Newton method is presented for solving the free-boundary problem composed of the velocity and pressure fields and the yield surfaces for creeping flow. Besides the outer surface, solid occurs as caps at the front and back of the sphere because of the stagnation points in the flow. The accuracy of solutions is ascertained by mesh refinement and by calculation of the integrals corresponding to the maximum and minimum variational principles for the problem. Large differences from the Newtonian values in the flow pattern around the sphere and in the drag coefficient are predicted, depending on the dimensionless value of the critical yield stressYgbelow which the material acts as a solid. The computed flow fields differ appreciably from Stokes’ solution. The sphere will fall only whenYgis below 0.143 For yield stresses near this value, a plastic boundary layer forms next to the sphere. Boundary-layer scalings give the correct forms of the dependence of the drag coefficient and mass-transfer coefficient on yield stress for values near the critical one. The Stokes limit of zero yield stress is singular in the sense that for any small value ofYgthere is a region of the flow away from the sphere where the plastic portion of the viscosity is at least as important as the Newtonian part. Calculations For the approach of the flow field to the Stokes result are in good agreement with the scalings derived from the matched asymptotic expansion valid in this limit.

Variational Field Theory

Equations which define a ``consistent'' set of ``boundary'' conditions, and hence a field, for a given set of differential equations are derived from a variational principle. The equivalence of functionals defined over an entire domain and functionals defined over only a subdomain, but with a surface term added to account for the contribution of the excluded subdomain, is exploited. The appropriate surface term is found to satisfy the Hamilton-Jacobi equation. The formalism is specialized to neutron diffusion theory, and it is demonstrated that Stark's double sweep method follows as a natural consequence of this field theoretic formulation. The relation of this formalism to Pontryagin's Maximum Principle and Bellman's Dynamic Programming is demonstrated for problems which can be characterized by minimum variational principles.

Time-Dependent Perturbation—Variation Method. III

It is shown that by a slight extension of the method previously presented one can use a minimum variational principle for time-dependent perturbational problems instead of the original stationarity principle.

Variational Principle for Potential Scattering

Potential scattering is treated by a minimum variational principle. The method, known as the error method, is applicable to problems which can be cast as linear inhomogeneous equations. The method makes use of a non-negative functional which reduces to zero for the exact solution. The magnitude of the functional for approximate solutions provides an indication of the accuracy of the wave function. The method is illustrated by some examples, and compared with the Schwinger and Kohn principles.