Asymmetric Design Sensitivity and Isogeometric Shape Optimization Subject to Deformation-Dependent Loads

We present a design sensitivity analysis and isogeometric shape optimization with path-dependent loads belonging to non-conservative loads under the assumption of elastic bodies. Path-dependent loads are sometimes expressed as the follower forces, and these loads have characteristics that depend not only on the design area of the structure but also on the deformation. When such a deformation-dependent load is considered, an asymmetric load stiffness matrix (tangential operator) in the response region appears. In this paper, the load stiffness matrix is derived by linearizing the non-linear non-conservative load, and the geometrical non-linear structure is optimally designed in the total Lagrangian formulation using the isogeometric framework. In particular, since the deformation-dependent load changes according to the change and displacement of the design area, the isogeometric analysis has a significant influence on the accuracy of the sensitivity analysis and optimization results. Through several numerical examples, the applicability and superiority of the isogeometric analysis method were verified in optimizing the shape of the problem subject to deformation-dependent loads.

Simple deformation measures for discrete elastic rods and ribbons

The discrete elastic rod method (Bergou et al. 2008 ACM Trans. Graph . 27 , 63:1–63:12. ( doi:10.1145/1360612.1360662 )) is a numerical method for simulating slender elastic bodies. It works by representing the centreline as a polygonal chain, attaching two perpendicular directors to each segment and defining discrete stretching, bending and twisting deformation measures and a discrete strain energy. Here, we investigate an alternative formulation of this model based on a simpler definition of the discrete deformation measures. Both formulations are equally consistent with the continuous rod model. Simple formulae for the first and second gradients of the discrete deformation measures are derived, making it easy to calculate the Hessian of the discrete strain energy. A few numerical illustrations are given. The approach is also extended to inextensible ribbons described by the Wunderlich model, and both the developability constraint and the dependence of the energy on the strain gradients are handled naturally.

ABOUT THE ANALYTICAL SOLUTION OF THE EQUATION OF IMPACT FORCE OF TWO ELASTIC BODIES

A nonlinear differential equation of the force of direct central impact of elastic bodies of revolution, which have a singular point on the boundary contact surface, where its curvature is infinite, is compiled. To determine the coefficients of the equation and the order of its power nonlinearity, the well-known solution of the axisymmetric contact problem of the theory of elasticity, constructed by I. Ya. Shtaermann, is used. In the formulation of the dynamic problem, the classical assumptions of the theory of quasi-static impact proposed by H. Hertz were also used. The constituted equation of impact force is reduced to the Bernoulli equation and its closed analytical solution is constructed, which is expressed in terms of the Ateb-sine. Analytical time dependences of the impact force and the convergence of the centers of mass of elastic bodies are obtained. Compact formulas have been derived for calculating the maxima of these quantities, as well as the durations of the process of compression and impact of bodies. Compact approximations of Ateb-sine by elementary functions are proposed. Thanks to these approximations, it was possible to obtain a fairly simple analytical sweep in time of a fast-flowing mechanical process. Traditionally, in other works such a scan was obtained by numerical solution of the corresponding integral equations that determine the force of an impact. Examples of calculations are given in which the influence of various factors on the main characteristics of a body impact with a small initial velocity is investigated. The limitation on the collision rate is due to the elastic formulation of the problem, where the possibility of plastic deformations is excluded. As a result of this formulation, the need to determine the rate of recovery rate has disappeared, for it is equal to one. Comparison of numerical results is carried out, to which the obtained analytical solutions and the numerical integration of the impact force equation on a computer lead. Small divergences of the results confirmed the accuracy of the derived calculation formulas. Numerical results relate to the impact of a steel body on a fixed rubber half-space, the analogue of which is observed in practice when falling pieces of mineral raw materials on the rolls of a vibration classifier lined with rubber.

Influence of Impulse Moments On Single-Frequency Torsional Oscillations of Nonlinearly Elastic Bodies

Analytical study of the impulse moment influences on the nonlinear torsional oscillations in the homogeneous constant cross-section of a body under classical boundary conditions of the first, second, and third types has been developed. For the case when the elastic material properties meet the body close to the power law of elasticity, mathematical models of the process are obtained. They are the boundary value problems for an equation of hyperbolic type with a small parameter at the discrete right-hand side. The latter expresses the effect of pulse momentum on the oscillatory process. The peculiarities of resonant oscillations are established. Relative torsional oscillations of a nonlinear elastic body that rotates around the axis with a constant portable angular velocity are considered, taking into account the periodic action of pulse momentum acting in a fixed cross-section. The reliability of the obtained calculation formulas is confirmed.