A new conservative/dissipative time integration scheme for nonlinear mechanical systems

Abstract

We present a conservative/dissipative time integration scheme for nonlinear mechanical systems. Starting from a weak form, we derive algorithmic forces and velocities that guarantee the desired conservation/dissipation properties. Our approach relies on a collection of linearly constrained quadratic programs defining high order correction terms that modify, in the minimum possible way, the classical midpoint rule so as to guarantee the strict energy conservation/dissipation properties. The solution of these programs provides explicit formulas for the algorithmic forces and velocities which can be easily incorporated into existing implementations. Similarities and differences between our approach and well-established methods are discussed as well. The approach, suitable for reduced-order models, finite element models, or multibody systems, is tested and its capabilities are illustrated by means of several examples.

Precision quotient

It is very useful to have means for checking the correctness of integration algorithms during their development and implementation. Therefore, we introduce here two tests that can be applied once an integration scheme has been numerically implemented. According to Kreiss and Ortiz [38], the numerical solution of an initial value problem can be expanded as

$$\begin{aligned} \varvec{\xi }(t,h,k)= & {} \varvec{\xi }(t)+ \left( \frac{h}{k}\right) \varvec{\psi }_{1}(t)+ \left( \frac{h}{k}\right) ^{2}\varvec{\psi }_{2}(t)+ \cdots \nonumber \\&+\left( \frac{h}{k}\right) ^{n}\varvec{\psi }_{n}(t)+ \mathcal {O}(h^{n+1}), \end{aligned}$$

(107)

where $\varvec{\xi }(t)$ is the exact solution of the given initial value problem and $\varvec{\psi }_{i}$ for $i=1,\ldots ,n$ are smooth functions of the time t that do not depend on the reference time step h. A positive integer number k allows to define finer solutions based on the original resolution given by the time step h that are necessary to compute precision coefficients, a tool that may be very effective to check the correctness of a running program.

A first precision quotient can be defined as

$$\begin{aligned} Q_{\text {I}}(t)=\frac{\left\| \varvec{\xi }(t,h,1)-\varvec{\xi }(t)\right\| }{\left\| \varvec{\xi }(t,h,2)-\varvec{\xi }(t)\right\| }, \end{aligned}$$

(108)

where the numerator is computed as

$$\begin{aligned} \left\| \varvec{\xi }(t,h,1)-\varvec{\xi }(t)\right\| =\left( \frac{h}{1}\right) ^{n}\left\| \varvec{\psi }_{n}(t)\right\| +\mathcal {O}(h^{n+1}) , \end{aligned}$$

(109)

and the denominator is given by

$$\begin{aligned} \left\| \varvec{\xi }(t,h,2)-\varvec{\xi }(t)\right\| =\left( \frac{h}{2}\right) ^{n}\left\| \varvec{\psi }_{n}(t)\right\| +\mathcal {O}(h^{n+1}). \end{aligned}$$

(110)

It is possible to show that for sufficiently small time steps, the first precision quotient can be directly approximated by $2^{n}$, where n denotes the order of accuracy of the integration method, namely

$$\begin{aligned} Q_{\text {I}}(t)=\frac{\left( \frac{h}{1}\right) ^{n}\left\| \varvec{\psi }_{n}(t)\right\| +\mathcal {O}(h^{n+1})}{\left( \frac{h}{2}\right) ^{n}\left\| \varvec{\psi }_{n}(t)\right\| +\mathcal {O}(h^{n+1})}=2^{n}+\mathcal {O}(h^{n+1})\approx 2^{n}. \end{aligned}$$

(111)

The main issue with this definition is that the exact solution of the initial value problem is required and, in general, is not available, especially in the context of mechanical systems involving nonlinear constitutive relations. To circumvent this drawback, it is possible to define a second precision quotient as

$$\begin{aligned} Q_{\text {II}}(t)=\frac{\left\| \varvec{\xi }(t,h,1)-\varvec{\xi }(t,h,2)\right\| }{\left\| \varvec{\xi }(t,h,2)-\varvec{\xi }(t,h,4)\right\| }, \end{aligned}$$

(112)

where the numerator is computed as

$$\begin{aligned} \begin{aligned}&\left\| \varvec{\xi }(t,h,1)-\varvec{\xi }(t,h,2)\right\| \\&\quad = \left\| \left( \frac{h}{1}\right) ^{n}\varvec{\psi }_{n}(t)-\left( \frac{h}{2}\right) ^{n}\varvec{\psi }_{n}(t)+\mathcal {O}(h^{n+1})\right\| \\&\quad = \left( \frac{2^{n}-1}{2^{n}}\right) h^{n}\left\| \varvec{\psi }_{n}(t)\right\| +\mathcal {O}(h^{n+1}) \end{aligned} \end{aligned}$$

(113)

and the denominator is given by

$$\begin{aligned} \begin{aligned}&\left\| \varvec{\xi }(t,h,2)-\varvec{\xi }(t,h,4)\right\| \\&\quad = \left\| \left( \frac{h}{2}\right) ^{n}\varvec{\psi }_{n}(t)-\left( \frac{h}{4}\right) ^{n}\varvec{\psi }_{n}(t)+\mathcal {O}(h^{n+1})\right\| \\&\quad = \left( \frac{2^{n}-1}{4^{n}}\right) h^{n}\left\| \varvec{\psi }_{n}(t)\right\| +\mathcal {O}(h^{n+1}). \end{aligned} \end{aligned}$$

(114)

Notice that this concept removes intrinsically the need for the exact solution of the considered initial value problem. Once again, it is possible to show that for sufficiently small time steps, the second precision quotient can be approximated by $2^{n}$ as well as in the case of the first precision quotient, namely

$$\begin{aligned} Q_{\text {II}}(t)= \frac{\left( \frac{2^{n}-1}{2^{n}}\right) h^{n}\left\| \varvec{\psi }_{n}(t)\right\| +\mathcal {O}(h^{n+1})}{\left( \frac{2^{n}-1}{4^{n}}\right) h^{n}\left\| \varvec{\psi }_{n}(t)\right\| +\mathcal {O}(h^{n+1})}= 2^{n}+\mathcal {O}(h^{n+1}) \approx 2^{n}. \end{aligned}$$

(115)

For the integration scheme considered in this work (an energy-conservative/dissipative method), accuracy of second order can be guaranteed, meaning that $\log _{2}[Q_{\text {I}}(t)]\approx 2$ and $\log _{2}[Q_{\text {II}}(t)]\approx 2$. Let us note that for the calculation of precision quotients, h has to be chosen small enough, and the choice may vary from case to case. In addition, if $\Vert \varvec{\psi }_{n}(t)\Vert$ is very small, both tests may fail even if the implementation is right. For this reason it is sometime necessary to experiment with several initial conditions and time step sizes in order to achieve correct pictures. As a general rule, the quotients of accuracy show better performance when the trajectories are periodic or quasi-periodic.