Narrow Results

In this study we analyzed the deformation of the polymeric rod impacting on the rigid wall which is called "Taylor impact test."" We simulated three-dimensional Taylor impact test depending on the various polymeric materials using the explicit finite element method by employing DYNA3D code. In simulation, polymeric materials were modeled using viscoelastic constitutive relations with the relaxation time and shear modulus. We have carried out the numerical simulation for the transient deformation characteristics and discussed effects of the viscoelastic constants on the deformation of the polymeric rod under impact.

Variational Integrator (VI) is a numerical technique, in which the Lagrangian of the system is used as the action integral. It is a special type of numerical solution that preserves the energy and momentum of the system. In this paper, we retrieve numerical solutions for heat and wave equation with the help of all possible combinations of finite difference scheme like forward–forward, forward–backward, forward–centered, backward–forward, backward–backward, backward–centered, centered–forward, centered–backward, centered–centered. We also use Lagrangian approach along with the projection technique to obtain approximate solutions of these linear models. This approach provides the best approximate solutions as well as preserves the energy of the system while the finite difference scheme gives only the numerical solutions. We also draw a comparison of existing exact solution with all approximate solutions for both models and provide graphical representation of these solutions.

It is shown in detail that Noether's theorem represents a mathematical identity which always exists for any nondegenerate functional and includes two parts: one is Euler–Lagrange expression, and the other is a possible conservation law. If and only if any physical equations are the same as Euler–Lagrange expressions, the following three points follow: (i) there always exist conservation laws or, at least, balance laws; (ii) the symmetry transformations between inertial frames can be used first to check the absolute invariance; (iii) the field variables included in spacetime transformations can be canceled for absolute invariance, and space and time transformations are irrelevant. A theorem for getting nonclassical conservation laws from the general solution of a physical system is presented. The necessary conditions in a theorem's form are also presented to construct Lagrangians for second- and fourth-order partial differential equations (PDEs). Based on these theorems, the Lagrangians and conservation laws of nonlinear heat and KdV equations are constructed and given. For Boussinesq's solution, a nonclassical conservation integral is also found for application. Moreover, it is shown that there exists only the same conservation laws derived from the generalized variational principle as those of the principle of minimum potential energy in elasticity.

Several applications of Lie symmetries and its generalisation are presented: from turning butterflies into tornados, to its applications in epidemics, population dynamics, and ultimately converting classical problems into the quantum realm. Applications of nonclassical symmetries are also illustrated.