Symbolic-Numeric Integration of the Dynamical Cosserat Equations

Abstract

We devise a symbolic-numeric approach to the integration of the dynamical part of the Cosserat equations, a system of nonlinear partial differential equations describing the mechanical behavior of slender structures, like fibers and rods. This is based on our previous results on the construction of a closed form general solution to the kinematic part of the Cosserat system. Our approach combines methods of numerical exponential integration and symbolic integration of the intermediate system of nonlinear ordinary differential equations describing the dynamics of one of the arbitrary vector-functions in the general solution of the kinematic part in terms of the module of the twist vector-function. We present an experimental comparison with the well-established generalized $\alpha$-method illustrating the computational efficiency of our approach for problems in structural mechanics.

A Generalized \(\alpha \)-Method

In this appendix, we briefly explain the application of the generalized $\alpha$-method for the common case of a system described by the standard equation from structural mechanics,

$$\begin{aligned} \varvec{M}\ddot{\varvec{x}}+\varvec{D}\dot{\varvec{x}}+\varvec{K}\varvec{x}+\varvec{\varLambda }(t)=\varvec{0}, \end{aligned}$$

(13)

in which $\varvec{M}$, $\varvec{D}$, and $\varvec{K}$ denote the mass, damping, and stiffness matrices. The time-dependent displacement vector is given by $\varvec{x}(t)$, and its first- and second-order temporal derivatives describe velocity and acceleration. The vector $\varvec{\varLambda }(t)$ describes external forces acting on the system at time t. We are searching for functions $\varvec{x}(t)$, $\varvec{\upsilon }(t)=\dot{\varvec{x}}(t)$, and $\varvec{a}(t)=\ddot{\varvec{x}}(t)$ satisfying (13) for all t with initial conditions $\varvec{x}(t_0)=\varvec{x}_0$ and $\varvec{\upsilon }(t_0)=\varvec{\upsilon }_0$.

For the employment of the generalized $\alpha$-method, we can write the integration scheme with respect to (13) as follows:

$$ \varvec{M}\varvec{a}_{1-\alpha _m}+\varvec{D}\varvec{x}_{1-\alpha _f}+\varvec{K}\varvec{x}_{1-\alpha _f}+\varvec{\varLambda }(t_{1-\alpha _f})=\varvec{0}, $$

with the substitution rule $(\cdot )_{1-\alpha }:=(1-\alpha )(\cdot )_i+\alpha (\cdot )_{i-1}$ and the approximations

$$\begin{aligned} \varvec{x}_i=\varvec{x}_{i-1}+\varDelta t\varvec{v}_{i-1}+\varDelta t^2\left( \left( \frac{1}{2}-\alpha \right) \varvec{a}_{i-1}+\beta \varvec{a}_i\right) ,\\ \varvec{\upsilon }_i=\varvec{\upsilon }_{i-1}+\varDelta t\left( \left( 1-\gamma \right) \varvec{a}_{i-1}+\gamma \varvec{a}_i\right) . \end{aligned}$$

The parameters $\alpha _m$, $\alpha _f$, $\gamma$, and $\beta$ are integration coefficients.