9.1 Introduction

We now give a brief introduction to time-dependent problems through the equations of elastodynamics for infinitesimal deformations

$$\begin{aligned} \nabla \cdot {\varvec{\sigma }}+{\varvec{f}}=\rho _o\frac{d^2{\varvec{u}}}{dt^2}=\rho _o\frac{d{\varvec{v}}}{dt}, \end{aligned}$$
(9.1)

where \(\nabla =\nabla _X\) and \(\frac{d}{dt}=\frac{\partial }{\partial t}\) (see Appendix B).

9.2 Generic Time Stepping

In order to motivate the time-stepping process, we first start with the dynamics of single point mass under the action of a force \({\varvec{\varPsi }}\). The equation of motion is given by (Newton’s Law)

$$\begin{aligned} m\dot{{\varvec{v}}}={\varvec{\varPsi }}, \end{aligned}$$
(9.2)

where \({\varvec{\varPsi }}\) is the total force applied to the particle. Expanding the velocity in a Taylor series about \(t+\theta \varDelta t\), where \(0\le \theta \le 1\), for \({\varvec{v}}(t+\varDelta t)\), we obtain

$$\begin{aligned} {\varvec{v}}(t+\varDelta t)={\varvec{v}}(t+\theta \varDelta t)+ \frac{d {\varvec{v}}}{d t}|_{t+\theta \varDelta t}(1-\theta )\varDelta t+ \frac{1}{2} \frac{d ^2 {\varvec{v}}}{d t^2}|_{t+\theta \varDelta t}(1-\theta )^2(\varDelta t)^2+\mathcal{O}(\varDelta t)^3 \end{aligned}$$
(9.3)

and for \({\varvec{v}}(t)\), we obtain

$$\begin{aligned} {\varvec{v}}(t)={\varvec{v}}(t+\theta \varDelta t) -\frac{d {\varvec{v}}}{d t}|_{t+\theta \varDelta t}\theta \varDelta t+ \frac{1}{2} \frac{d ^2 {\varvec{v}}}{d t^2}|_{t+\theta \varDelta t}\theta ^2(\varDelta t)^2+\mathcal{O}(\varDelta t)^3. \end{aligned}$$
(9.4)

Subtracting the two expressions yields

$$\begin{aligned} \frac{d {\varvec{v}}}{d t}|_{t+\theta \varDelta t} =\frac{{\varvec{v}}(t+\varDelta t)-{\varvec{v}}(t)}{\varDelta t}+\hat{\mathcal{O}}(\varDelta t), \end{aligned}$$
(9.5)

where \(\hat{\mathcal{O}}(\varDelta t)=\mathcal{O}(\varDelta t)^2\), when \(\theta =\frac{1}{2}\), otherwise \(\hat{\mathcal{O}}(\varDelta t)=\mathcal{O}(\varDelta t)\). Thus, inserting this into Eq. 9.2 yields

$$\begin{aligned} {\varvec{v}}(t+\varDelta t)={\varvec{v}}(t)+\frac{\varDelta t}{m}{\varvec{\varPsi }}(t+\theta \varDelta t)+\hat{\mathcal{O}}(\varDelta t)^2. \end{aligned}$$
(9.6)

Note that a weighted sum of Eqs. 9.3 and 9.4 yields

$$\begin{aligned} {\varvec{v}}(t+\theta \varDelta t)=\theta {\varvec{v}}(t+\varDelta t)+(1-\theta ) {\varvec{v}}(t) +\mathcal{O}(\varDelta t)^2, \end{aligned}$$
(9.7)

which will be useful shortly. Now expanding the position of the mass in a Taylor series about \(t+\theta \varDelta t\) we obtain

$$\begin{aligned} {\varvec{u}}(t+\varDelta t)={\varvec{u}}(t+\theta \varDelta t)+ \frac{d {\varvec{u}}}{d t}|_{t+\theta \varDelta t}(1-\theta )\varDelta t+ \frac{1}{2} \frac{d ^2 {\varvec{u}}}{d t^2}|_{t+\theta \varDelta t}(1-\theta )^2(\varDelta t)^2+\mathcal{O}(\varDelta t)^3 \end{aligned}$$
(9.8)

and

$$\begin{aligned} {\varvec{u}}(t)={\varvec{u}}(t+\theta \varDelta t) -\frac{d {\varvec{u}}}{d t}|_{t+\theta \varDelta t}\theta \varDelta t+ \frac{1}{2} \frac{d ^2 {\varvec{u}}}{d t^2}|_{t+\theta \varDelta t}\theta ^2(\varDelta t)^2+\mathcal{O}(\varDelta t)^3. \end{aligned}$$
(9.9)

Subtracting the two expressions yields

$$\begin{aligned} \frac{{\varvec{u}}(t+\varDelta t)-{\varvec{u}}(t)}{\varDelta t}={\varvec{v}}(t+\theta \varDelta t)+\hat{\mathcal{O}}(\varDelta t). \end{aligned}$$
(9.10)

Inserting Eq. 9.7 yields

$$\begin{aligned} {\varvec{u}}(t+\varDelta t)={\varvec{u}}(t)+\left( \theta {\varvec{v}}(t+\varDelta t)+(1-\theta ) {\varvec{v}}(t)\right) \varDelta t +\hat{\mathcal{O}}(\varDelta t)^2, \end{aligned}$$
(9.11)

and using Eq. 9.6 yields

$$\begin{aligned} {\varvec{u}}(t+\varDelta t)={\varvec{u}}(t)+{\varvec{v}}(t)\varDelta t+\frac{\theta (\varDelta t)^2}{m}{\varvec{\varPsi }}(t+\theta \varDelta t)+\hat{\mathcal{O}}(\varDelta t)^2. \end{aligned}$$
(9.12)

The term \({\varvec{\varPsi }}(t+\theta \varDelta t)\) can be handled in a simple way:

$$\begin{aligned} {\varvec{\varPsi }}(t+\theta \varDelta t)\approx \theta {\varvec{\varPsi }}(t+\varDelta t)+(1-\theta ){\varvec{\varPsi }}(t). \end{aligned}$$
(9.13)

We note that

  • When \(\theta =1\), then this is the (implicit) Backward Euler scheme, which is very stable (very dissipative) and \(\hat{\mathcal{O}}(\varDelta t)^2=\mathcal{O}(\varDelta t)^2\) locally in time,

  • When \(\theta =0\), then this is the (explicit) Forward Euler scheme, which is conditionally stable and \(\hat{\mathcal{O}}(\varDelta t)^2=\mathcal{O}(\varDelta t)^2\) locally in time,

  • When \(\theta =0.5\), then this is the (implicit) “Midpoint” scheme, which is stable and \(\hat{\mathcal{O}}(\varDelta t)^2=\mathcal{O}(\varDelta t)^3\) locally in time.

In summary, we have for the velocityFootnote 1

$$\begin{aligned} {\varvec{v}}(t+\varDelta t)={\varvec{v}}(t)+\frac{\varDelta t}{m} \left( \theta {\varvec{\varPsi }}(t+\varDelta t) +(1-\theta ){\varvec{\varPsi }}(t)\right) \end{aligned}$$
(9.14)

and for the position

$$\begin{aligned} {\varvec{u}}(t+\varDelta t)= & {} {\varvec{u}}(t)+{\varvec{v}}(t+\theta \varDelta t)\varDelta t \\= & {} {\varvec{u}}(t)+\left( \theta {\varvec{v}}(t+\varDelta t)+(1-\theta ) bfv(t)\right) \varDelta t, \nonumber \end{aligned}$$
(9.15)

or in terms of \({\varvec{\varPsi }}\)

$$\begin{aligned} {\varvec{u}}(t+\varDelta t)={\varvec{u}}(t)+{\varvec{v}}(t)\varDelta t +\frac{\theta (\varDelta t)^2}{m} \left( \theta {\varvec{\varPsi }}(t+\varDelta t)+(1-\theta ){\varvec{\varPsi }}(t)\right) . \end{aligned}$$
(9.16)

9.3 Application to the Continuum Formulation

Now consider the continuum analogue to “\(m\dot{{\varvec{v}}}\)”

$$\begin{aligned} \rho _o\frac{\partial ^2{\varvec{u}}}{\partial t^2}=\rho _o\frac{\partial {\varvec{v}}}{\partial t} =\nabla \cdot {\varvec{\sigma }}+{\varvec{f}}{\mathop {=}\limits ^{\mathrm{def}}}{\varvec{\varPsi }}\end{aligned}$$
(9.17)

and thus

$$\begin{aligned} \rho _o{\varvec{v}}(t+\varDelta t)=\rho _o{\varvec{v}}(t)+\varDelta t \left( \theta {\varvec{\varPsi }}(t+\varDelta t)+(1-\theta ){\varvec{\varPsi }}(t)\right) . \end{aligned}$$
(9.18)

Multiplying Eq. 9.18 by a test function and integrating yields

$$\begin{aligned} \int _{\varOmega }{\varvec{\nu }}\cdot \rho _o{\varvec{v}}(t+\varDelta t)\, d\varOmega= & {} \int _{\varOmega }{\varvec{\nu }}\cdot \rho _o{\varvec{v}}(t)\, d\varOmega \\+ & {} \varDelta t\int _{\varOmega }{\varvec{\nu }}\cdot \left( \theta {\varvec{\varPsi }}(t+\varDelta t)+(1-\theta ){\varvec{\varPsi }}(t)\right) \, d\varOmega , \nonumber \end{aligned}$$
(9.19)

and using Gauss’s divergence theorem and enforcing \({\varvec{\nu }}=\mathbf {0}\) on \(\varGamma _u\) yields (using a streamlined time-step superscript counter notation of L, where \(t=L\varDelta t\) and \(t+\varDelta t=(L+1)\varDelta t\))

$$\begin{aligned} \int _{\varOmega }{\varvec{\nu }}\cdot \rho _o{\varvec{v}}^{L+1}\, d\varOmega= & {} \int _{\varOmega }{\varvec{\nu }}\cdot \rho _o{\varvec{v}}^L\, d\varOmega \\+ & {} \varDelta t \theta \left( -\int _{\varOmega }\nabla {\varvec{\nu }}:{\varvec{\sigma }}\,d\varOmega +\int _{\varGamma _t}{\varvec{\nu }}\cdot ({\varvec{\sigma }}\cdot {\varvec{n}}) \,dA +\int _{\varOmega }{\varvec{\nu }}\cdot {\varvec{f}}\,d\varOmega \right) ^{L+1} \nonumber \\+ & {} \varDelta t(1-\theta ) \left( -\int _{\varOmega }\nabla {\varvec{\nu }}:{\varvec{\sigma }}\,d\varOmega +\int _{\varGamma _t}{\varvec{\nu }}\cdot {\varvec{t}}^* \,dA +\int _{\varOmega }{\varvec{\nu }}\cdot {\varvec{f}}\,d\varOmega \right) ^{L}.\nonumber \end{aligned}$$
(9.20)

As in the previous chapter on linearized three-dimensional elasticity, we assume

$$\begin{aligned} \{{\varvec{u}}^h\}=[{\varvec{\varPhi }}]\{{\varvec{a}}\} \qquad and \qquad \{{\varvec{\nu }}^h\}=[{\varvec{\varPhi }}]\{{\varvec{b}}\} \qquad and \qquad \{{\varvec{v}}^h\}=[{\varvec{\varPhi }}]\{\dot{{\varvec{a}}}\}, \end{aligned}$$
(9.21)

which yields, in terms of matrices and vectors

$$\begin{aligned} \{{\varvec{b}}\}^T[M]\{\dot{{\varvec{a}}}\}^{L+1}= & {} \{{\varvec{b}}\}^T[M]\{\dot{{\varvec{a}}}\}^{L} -\varDelta t \theta \{{\varvec{b}}\}^T \left( -[K]\{{\varvec{a}}\}^{L+1}+\{{\varvec{R}}_f\}^{L+1}+\{{\varvec{R}}_t\}^{L+1}\right) \nonumber \\- & {} \{{\varvec{b}}\}^T\varDelta t (1-\theta ) \left( -[K]\{{\varvec{a}}\}^{L}+\{{\varvec{R}}_f\}^{L}+\{{\varvec{R}}_t\}^{L}\right) . \end{aligned}$$
(9.22)

where \([M]=\int _{\varOmega }\rho _o[\Phi ]^T[\Phi ]\, d\varOmega \), and [K],\(\{{\varvec{R}}_f\}\), and \(\{{\varvec{R}}_t\}\) are as defined in the previous chapters on elastostatics. Note that \(\{{\varvec{R}}_f\}^{L}\) and \(\{{\varvec{R}}_t\}^{L}\) are known values from the previous time-step. Since \(\{{\varvec{b}}\}^T\) is arbitrary

$$\begin{aligned}{}[M]\{\dot{{\varvec{a}}}\}^{L+1}= & {} [M]\{\dot{{\varvec{a}}}\}^{L}+(\varDelta t \theta ) \left( -[K]\{{\varvec{a}}\}^{L+1}+\{{\varvec{R}}_f\}^{L+1}+\{{\varvec{R}}_t\}^{L+1}\right) \nonumber \\+ & {} \varDelta t (1-\theta ) \left( -[K]\{{\varvec{a}}\}^{L}+\{{\varvec{R}}_f\}^{L}+\{{\varvec{R}}_t\}^{L}\right) . \end{aligned}$$
(9.23)

One should augment this with the approximation for the discrete displacement:

$$\begin{aligned} \{{\varvec{a}}\}^{L+1}=\{{\varvec{a}}\}^{L}+\varDelta t\left( \theta \{\dot{{\varvec{a}}}\}^{L+1}+(1-\theta )\{\dot{{\varvec{a}}}\}^{L}\right) . \end{aligned}$$
(9.24)

For a purely implicit (Backward Euler) method \(\theta =1\)

$$\begin{aligned} \left( [M]\{\dot{{\varvec{a}}}\}^{L+1}+\varDelta t[K]\{{\varvec{a}}\}^{L+1}\right) =[M]\{\dot{{\varvec{a}}}\}^{L}+\varDelta t\left( \{{\varvec{R}}_t\}^{L+1}+\{{\varvec{R}}_f\}^{L+1}\right) , \end{aligned}$$
(9.25)

augmented with

$$\begin{aligned} \{{\varvec{a}}\}^{L+1}=\{{\varvec{a}}\}^{L}+\varDelta t\{\dot{{\varvec{a}}}\}^{L+1}, \end{aligned}$$
(9.26)

which requires one to solve a system of algebraic equations, while for an explicit (Forward Euler) method \(\theta =0\) with usually [M] is approximated by an easy-to-invert matrix, such as a diagonal matrix, \([M]\approx M[\mathbf{1}]\), to make the matrix inversion easy, yielding:

$$\begin{aligned} \{\dot{{\varvec{a}}}\}^{L+1}=\{\dot{{\varvec{a}}}\}^{L}+\varDelta t[M]^{-1}\left( -[K]\{{\varvec{a}}\}^{L}+\{{\varvec{R}}_f\}^{L}+\{{\varvec{R}}_t\}^{L}\right) , \end{aligned}$$
(9.27)

augmented with

$$\begin{aligned} \{{\varvec{a}}\}^{L+1}=\{{\varvec{a}}\}^{L}+\varDelta t\{\dot{{\varvec{a}}}\}^{L}. \end{aligned}$$
(9.28)

There is an enormous number of time-stepping schemes. For general time-stepping, we refer the reader to the seminal texts of Hairer et al. [1, 2]. In the finite element context, we refer the reader to Bathe [3], Becker et al. [4], Hughes [5], and Zienkiewicz and Taylor [6].