Vadim Komkov

Sensitivity techniques for systems with distributed parameters

Abstract We derive a simple sensitivity formula for systems described by the equation Lw = q where L is a partial differential operator that depends on a vector of parameters h. The applications of the formula suggests that different functional analytic settings may offer computational difficulties and analytical advantages and vice versa.

Variational Principles of Continuum Mechanics

Let V be a linear vector space over the complex numbers, such that for any pair of vectors f, g, ɛ V there exists a complex number and the maps VxV→C called the inner (or scalar) product ·{f,g} → satisfies the following axioms i) \( \left\langle {f,g} \right\rangle = \left\langle {\overline {g,f} } \right. \) (bar denotes complex conjugate) ii) for any \( {{a}_{1}},{{a}_{2}} \in C,{{f}_{1}},{{f}_{2}},g \in V \). iii) 〈f,f〉 is a nonnegative real number (for any f ∈ V) and 〈f,f〉 \( = \Leftrightarrow f = \emptyset . \). iv) It follows from i) and iii) that \( \left\langle {af,g} \right\rangle = a\left\langle {f,g} \right\rangle = \left\langle {f,\bar{a}g} \right\rangle . \).

Euler's buckling formula and Wirtinger's inequality

This article contains a discussion of Eulers buckling formula for a compressed elastic column. The most commonly used classroom derivations of this formula follow roughly the original arguments of Euler. For this reason the history of this problem is briefly reviewed. Alternative methods of derivation use energy arguments, including Rayleighs principle and Kirchhofs energy inequality. Here we show that the quadratic strain energy formula can be used directly, in conjunction with the so called Wirtinger‐Poincare‐Almansi inequality to offer an extremely simple proof of the Eulers buckling load formula: Pcritilal = π2.EI/l2. To the best of our knowledge this derivation is new.

Design optimization for structural or mechanical systems against the worst possible loads

The main objective of this paper is to point out several difficulties related to formulating and solving numerically the problem of optimal design of structural systems subjected to ‘worst admissible’ controls (that is ‘worst’ external loads). First some known facts and available results are reviewed and minor lemmas are provided so that the problem can be formulated in an appropriate mathematical setting. In the second part of the paper numerical techniques including some algorithms are discussed. Convergence and proper numerical approaches to suboptimal designs are the main topics of this part. While the main concern is structural analysis, a short digression indicates that the techniques and arguments offered here are easily extended to other applications such as the mechanical and electro-magnetic systems design.

Energy Methods, Classical Calculus of Variations Approach — Selected Topics and Applications

The points of view of Hamilton, Lagrange, Gauss, Hertz, and Lord Rayleigh emphasized the concept of energy rather than force. The equations of motion of the system are not derived by consideration of equilibrium of forces acting on the system, possibly including the Newtonian and d’Alambert inertia forces. Instead, the primary role is played by energy considerations. As an example of such an approach, a condition of stable equilibrium under static loads is replaced by the condition of a local minimum for the potential energy of a mechanical system. Instead of solving the equations of motion of a vibrating system to find its natural frequencies, it is possible to consider the mean values of potential and kinetic energies, or to minimize an appropriate energy functional.

The Legendre Transformation, Duality and Functional Analytic Approach

The growth of science is accompanied by a steady diversification of knowledge. At the same time a deeper understanding of diverse phenomena leads to discoveries of common laws of physics and of mathematics, unifying the mathematical treatment. Similarity between inertial and gravitational forces lies at the origins of Einstein’s “thought experiments” and the special theory of relativity. Mach’s explanation of inertial forces in Newton’s bucket experiment is an example of a brilliant unification attempt. Basically, E. Mach suggested that the mass of the entire universe affects the local laws of mechanics so far discovered by man-kind. (We refer to his famous statement about the distance stars causing it all.)

Variational Formulation of Problems of Elastic Stability. Topics in the Stability of Beams and Plates

Several authors studying the theory of boundary value problems describing deformation, buckling, and steady state vibration of elastic systems have developed a variational approach based on functional analytic arguments. See Aubin [1]. Generally a system of equations of the classical boundary value type is reduced to a variational, or to an energy formulation. This is not a new technique. A rigorous existence and uniqueness theory can be developed in a formulation that provides a direct link with the engineering theory of finite element analysis. See Aubin [1], or Ciarlet [2].

Some Known Variational Principles In Elasticity

In the 1970-s and 80-s a significant change of style took place in variational treatment of continuum mechanics.

An Example of a Variational Approach to Mechanics Using “Different” Rules of Algebra

In many instances duality does not help in deriving complementary variational principles due to “wrong” signs of abstract second derivatives. The Lagrangian formulation “works”, the Hamilton’s canonical equations are displayed, everything seems to be in the right form. Upon a closer examination only one variational principle can be formulated at best. Even if the signs of the second FrSchet derivatives can be determined in the vicinity of a critical point of the cost functional, both are of the same sign. Suppose that the Lagrangian for the problem L,(x,Ax) vanishes at x = x.

A note on the behavior of solids under very high pressures

In this note we discuss the flow of certain solids such as annealed copper subjected to a pressure in the one megabar range. We consider the process of bubble formation and a numerical approach to this problem.ZusammenfassungWir diskutieren hier den Fluß gewisser Festkörper, wie z.B. von ausgeglühtem Kupfer, das einem Druck im 1 Megabar-Bereich unterworfen wird. Der Prozess der Blasenformation und eine numerische Annäherung dieses Problems werden betrachtet.