Goal-oriented space-time adaptivity for transient dynamics using a modal description of the adjoint solution

Abstract

This article presents a space-time adaptive strategy for transient elastodynamics. The method aims at computing an optimal space-time discretization such that the computed solution has an error in the quantity of interest below a user-defined tolerance. The methodology is based on a goal-oriented error estimate that requires accounting for an auxiliary adjoint problem. The major novelty of this paper is using modal analysis to obtain a proper approximation of the adjoint solution. The idea of using a modal-based description was introduced in a previous work for error estimation purposes. Here this approach is used for the first time in the context of adaptivity. With respect to the standard direct time-integration methods, the modal solution of the adjoint problem is highly competitive in terms of computational effort and memory requirements. The performance of the proposed strategy is tested in two numerical examples. The two examples are selected to be representative of different wave propagation phenomena, one being a 2D bulky continuum and the second a 2D domain representing a structural frame.

Linear system to be solved at each time step

This appendix details how the time-continuous Galerkin approximation is computed when the space mesh changes between times slabs.

Recall that the numerical approximation $\tilde{\mathbf{U}}$ solution of the discrete problem (5) is computed sequentially starting from the first time slab $I_1$ until the last one $I_N$. Specifically, assuming that the solution at the time-slab $I_{n-1}$ is known, the approximation $\tilde{\mathbf{U}}$ restricted to the slab $I_n$ is found solving the problem: find $\tilde{\mathbf{U}}|_{I_n}\in \varvec{\mathcal {W}}^{H,\Delta t}_u|_{I_n}\times \varvec{\mathcal {W}}^{H,\Delta t}_v|_{I_n}$ such that

$$\begin{aligned}&\int _{I_n}\left[ m({\dot{\tilde{\mathbf{u}}}}_v + a_1{\tilde{\mathbf{u}}}_v,\mathbf{w}_v) + a(\tilde{\mathbf{u}}_u+a_2\tilde{\mathbf{u}}_v,\mathbf{w}_v)\right] \mathrm {\ d}t \nonumber \\&\quad =\int _{I_n}l(t;\mathbf{w}_v) \mathrm {\ d}t,\quad \forall \mathbf{w}_v\in \varvec{\mathcal {V}}_0^{H}(\mathcal {P}_n),\end{aligned}$$

(33a)

$$\begin{aligned}&\int _{I_n} a({\dot{\tilde{\mathbf{u}}}}_u - \tilde{\mathbf{u}}_v,\mathbf{w}_u) \mathrm {\ d}t = 0,\quad \forall \mathbf{w}_u\in \varvec{\mathcal {V}}_0^{H}(\mathcal {P}_n),\end{aligned}$$

(33b)

$$\begin{aligned}&\tilde{\mathbf{U}}(t^+_{n-1}) = \tilde{\mathbf{U}}(t_{n-1}), \end{aligned}$$

(33c)

where, for $n>1$, $\tilde{\mathbf{U}}(t_{n-1})$ is the solution at the end of the previous interval $I_{n-1}$ and, for $n=1$, $\tilde{\mathbf{U}}(t_{n-1}=t_0)$ is defined using the initial conditions, $\tilde{\mathbf{U}}(t_0)=[\mathbf{u}_0,\mathbf{v}_0]$.

From the definition of the discrete spaces $\varvec{\mathcal {W}}^{H,\Delta t}_u$ and $\varvec{\mathcal {W}}^{H,\Delta t}_v$, the numerical displacements and velocities $\tilde{\mathbf{u}}_u$ and $\tilde{\mathbf{u}}_v$ inside the interval $I_n$ are expressed as a combination of the values at times $t_{n-1}$ and $t_n$, namely

$$\begin{aligned} \tilde{\mathbf{u}}_u|_{I_n}=\tilde{\mathbf{u}}_u(t_{n-1})\theta _{n-1}(t) + \tilde{\mathbf{u}}_u(t_n)\theta _n(t),\end{aligned}$$

(34a)

$$\begin{aligned} \tilde{\mathbf{u}}_v|_{I_n}=\tilde{\mathbf{u}}_v(t_{n-1})\theta _{n-1}(t) + \tilde{\mathbf{u}}_v(t_n)\theta _n(t). \end{aligned}$$

(34b)

Thus, using the initial conditions for the interval (33c), the values $\tilde{\mathbf{u}}_u(t_{n-1})$ and $\tilde{\mathbf{u}}_v(t_{n-1}) \in \varvec{\mathcal {V}}_0^{H}(\mathcal {P}_{n-1})mu$ are known and the only unknowns to be determined are $\tilde{\mathbf{u}}_u(t_{n})$ and $\tilde{\mathbf{u}}_v(t_{n}) \in \varvec{\mathcal {V}}_0^{H}(\mathcal {P}_n)$. These unknowns are found inserting the representation(34) in equation (33) and noting that the following properties of the time-shape functions hold,

$$\begin{aligned}&\int _{I_n} \theta _{n-1}(t) \mathrm {\ d}t = \int _{I_n} \theta _{n}(t) \mathrm {\ d}t = \frac{\Delta t_n}{2} \quad \text { and }\\&\quad -\int _{I_n} \dot{\theta }_{n-1}(t) \mathrm {\ d}t = \int _{I_n} \dot{\theta }_{n}(t) \mathrm {\ d}t = 1. \end{aligned}$$

Specifically, $[\tilde{\mathbf{u}}_u(t_n),\tilde{\mathbf{u}}_v(t_n)]\in \varvec{\mathcal {V}}_0^{H}(\mathcal {P}_n)\times \varvec{\mathcal {V}}_0^{H}(\mathcal {P}_n)$ is such that

$$\begin{aligned}&m(\tilde{\mathbf{u}}_v(t_n),\mathbf{w}_v) +\dfrac{\Delta t_n}{2}c(\tilde{\mathbf{u}}_v(t_n) ,\mathbf{w}_v) + \dfrac{\Delta t_n}{2}a(\tilde{\mathbf{u}}_u(t_n) ,\mathbf{w}_v) \nonumber \\&\quad = l_{v,n}(\mathbf{w}_v),\ \forall \mathbf{w}_v\in \varvec{\mathcal {V}}_0^{H}(\mathcal {P}_n), \end{aligned}$$

(35a)

and

$$\begin{aligned} a(\tilde{\mathbf{u}}_u(t_n),\mathbf{w}_u) - \dfrac{\Delta t_n}{2}a(\tilde{\mathbf{u}}_v(t_n),\mathbf{w}_u)\nonumber \\ =l_{u,n}(\mathbf{w}_u),\forall \mathbf{w}_u\in \varvec{\mathcal {V}}_0^{H}(\mathcal {P}_n), \end{aligned}$$

(35b)

where

$$\begin{aligned} l_{v,n}(\mathbf{w}) :&= \int _{I_n} l(t;\mathbf{w}) \mathrm {\ d}t + m(\tilde{\mathbf{u}}_v(t_{n-1}),\mathbf{w}) \\&- \dfrac{\Delta t_n}{2}c(\tilde{\mathbf{u}}_v(t_{n-1}) ,\mathbf{w}) - \dfrac{\Delta t_n}{2}a(\tilde{\mathbf{u}}_u(t_{n-1}) ,\mathbf{w}),\\ l_{u,n}(\mathbf{w}):&= a(\tilde{\mathbf{u}}_u(t_{n-1}),\mathbf{w}) + \dfrac{\Delta t_n}{2}a( \tilde{\mathbf{u}}_v(t_{n-1}),\mathbf{w}),\\ c(\mathbf{v},\mathbf{w}):&= m(a_1\mathbf{v},\mathbf{w}) + a(a_2\mathbf{v},\mathbf{w}). \end{aligned}$$

Note that since the values $\tilde{\mathbf{u}}_u(t_{n-1})$ and $\tilde{\mathbf{u}}_v(t_{n-1})$ are known, the terms associated with this values are placed in the right hand side of the equations.

The computation of the terms appearing in the left hand side of (35) entails no difficulty since all the spatial functions belong to $\varvec{\mathcal {V}}_0^{H}(\mathcal {P}_n)$. On the contrary, if different spatial computational meshes are used at times $t_{n-1}$ and $t_n$, the computation of the nodal force vectors associated with $l_{u,n}(\cdot )$ and $l_{v,n}(\cdot )$ involves computing mass and energy products of functions defined in the mesh at time $t_{n-1}$ and functions defined in the mesh at time $t_{n}$, e.g. $m(\tilde{\mathbf{u}}_v(t_{n-1}),\mathbf{w}_v)$.

The use of different spatial meshes is efficiently handled by solving the discrete problem (35) using the auxiliary union mesh $\mathcal {P}_{n-1,n}$ containing in each zone of the domain the finer elements either in $\mathcal {P}_{n-1}$ or $\mathcal {P}_{n}$, see Fig. 18, namely

$$\begin{aligned} \mathcal {P}_{n-1,n}:=\{ \omega = \triangle \cap \triangle ^\prime \text { for } \triangle \in \mathcal {P}_{n-1},\ \triangle ^\prime \in \mathcal {P}_{n}\}. \end{aligned}$$

Fig. 18

Illustration of the computational meshes $\mathcal {P}_{n-1},\mathcal {P}_{n}$, and their union $\mathcal {P}_{n-1,n}$

Note that, any function belonging either to $\varvec{\mathcal {V}}_0^{H}(\mathcal {P}_{n-1})$ or $\varvec{\mathcal {V}}_0^{H}(\mathcal {P}_n)$ can be represented in the finite element space associated to $\mathcal {P}_{n-1,n}$, namely $\varvec{\mathcal {V}}^H_0(\mathcal {P}_{n-1,n})$, without lose of information. Thus, the products involving functions in different meshes are efficiently computed after projecting the functions in the space $\varvec{\mathcal {V}}^H_0(\mathcal {P}_{n-1,n})$. However, discretizing problem (35) using the mesh $\mathcal {P}_{n-1,n}$ requires introducing additional constrains to enforce that the computed fields $\tilde{\mathbf{u}}_u(t_n)$ and $\tilde{\mathbf{u}}_v(t_n)$ belong to $\varvec{\mathcal {V}}_0^{H}(\mathcal {P}_n)$ and not to $\varvec{\mathcal {V}}^H_0(\mathcal {P}_{n-1,n})$. That is, problem (35) leads to the following system of equations when discretized in the auxiliary finite element mesh $\mathcal {P}_{n-1,n}$:

$$\begin{aligned} \left[ \begin{array}{llll} \mathbf{K}_n &{} -\frac{\Delta t_n}{2} \mathbf{K}_n &{} \mathbf{A}^\mathrm {T}_n &{} \mathbf {0}\\ \frac{\Delta t_n}{2} \mathbf{K}_n &{} \mathbf{M}_n + \frac{\Delta t_n}{2} \mathbf{C}_n &{} \mathbf {0} &{} \mathbf{A}^\mathrm {T}_n\\ \mathbf{A}_n &{} \mathbf {0} &{} \mathbf {0} &{} \mathbf {0}\\ \mathbf {0} &{} \mathbf{A}_n &{} \mathbf {0} &{} \mathbf {0} \end{array}\right] \left[ \begin{array}{l} \mathbf{U}_{u,n}\\ \mathbf{U}_{v,n}\\ \varvec{\lambda }_{u,n}\\ \varvec{\lambda }_{v,n} \end{array}\right] = \left[ \begin{array}{l} \mathbf{F}_{u,n} \\ \mathbf{F}_{v,n} \\ \mathbf {0} \\ \mathbf {0} \end{array}\right] \end{aligned}$$

(36)

where

$$\begin{aligned} \mathbf{F}_{u,n} :&= \mathbf{K}_n \mathbf{U}_{u,n-1} + \frac{\Delta t_n}{2} \mathbf{K}_n\mathbf{U}_{v,n-1},\\ \mathbf{F}_{v,n} :&= (\mathbf{M}_n- \dfrac{\Delta t_n}{2}\mathbf{C}_n)\mathbf{U}_{v,n-1} \\&- \dfrac{\Delta t_n}{2} \mathbf{K}_n\mathbf{U}_{u,n-1} + \int _{I_n} \mathbf{F}(t)\mathrm {\ d}t, \end{aligned}$$

and $\mathbf{C}_n := a_1 \mathbf{M}_n + a_2\mathbf{K}_n$. The matrices $\mathbf{M}_n$ and $\mathbf{K}_n$ and the vector $\mathbf{F}(t)$ are the discrete counterparts of the bilinear forms $m(\cdot ,\cdot )$ and $a(\cdot ,\cdot )$ and the linear form $l(t;\cdot )$ in the space $\varvec{\mathcal {V}}^H_0(\mathcal {P}_{n-1,n})$ and the vectors $\mathbf{U}_{u,n}$, $\mathbf{U}_{v,n}$, $\mathbf{U}_{u,n-1}$ and $\mathbf{U}_{v,n-1}$ contain the degrees of freedom of functions $\tilde{\mathbf{u}}_u(t_n)$, $\tilde{\mathbf{u}}_v(t_n)$, $\tilde{\mathbf{u}}_u(t_{n-1})$ and $\tilde{\mathbf{u}}_v(t_{n-1})$ expressed in the discrete space $\varvec{\mathcal {V}}^H_0(\mathcal {P}_{n-1,n})$. Note that the linear constrains $\mathbf{A}_n\mathbf{U}_{u,n}=\mathbf {0}$ and $\mathbf{A}_n\mathbf{U}_{v,n}=\mathbf {0}$ are introduced in order to ensure that the computed fields $\tilde{\mathbf{u}}_u(t_n)$ and $\tilde{\mathbf{u}}_u(t_n)$ belong to $\varvec{\mathcal {V}}_0^{H}(\mathcal {P}_n)$ and also to impose continuity of the solution at the hanging nodes, see Fig. 19. The vectors $\varvec{\lambda }_{u,n}$ and $\varvec{\lambda }_{v,n}$ are the associated Lagrange multipliers.

Fig. 19

The numerical solution is constrained at the nodes of the mesh $\mathcal {P}_{n-1,n}$ corresponding to hanging nodes in the mesh $\mathcal {P}_{n}$ and also at the nodes of $\mathcal {P}_{n-1,n}$ which disappear in mesh $\mathcal {P}_{n}$

Note that system (36) is at the first sight of double size than the one associated with the Newmark method. However, system (36) can be rewritten in a more convenient way by subtracting to the second row of the matrix in (36) the first row multiplied by $\frac{\Delta t_n}{2}$. That is,

$$\begin{aligned}&\left[ \begin{array}{llll} \mathbf{K}_n &{} -\frac{\Delta t_n}{2}\mathbf{K}_n &{} \mathbf{A}_n^\mathrm {T} &{} \mathbf {0}\\ \mathbf {0} &{} \mathbf{M}_n + \frac{\Delta t_n}{2}\mathbf{C}_n +\frac{\Delta t_n^2}{4}\mathbf{K}_n &{} - \frac{\Delta t_n}{2}\mathbf{A}_n^\mathrm {T} &{} \mathbf{A}_n^\mathrm {T}\\ \mathbf{A}_n &{} \mathbf {0} &{} \mathbf {0} &{} \mathbf {0}\\ \mathbf {0} &{} \mathbf{A}_n &{}\mathbf {0} &{} \mathbf {0} \end{array}\right] \left[ \begin{array}{l} \mathbf{U}_{u,n}\\ \mathbf{U}_{v,n}\\ \varvec{\lambda }_{u,n}\\ \varvec{\lambda }_{v,n} \end{array}\right] \\&\quad = \left[ \begin{array}{l} \mathbf{F}_{u,n} \\ \mathbf{F}_{v,n}-\frac{\Delta t_n}{2}\mathbf{F}_{u,n} \\ \mathbf {0} \\ \mathbf {0} \end{array}\right] \end{aligned}$$

This reformulation allows to compute the velocities separately from the displacements solving a system of the same size as the usual system arising in the Newmark method, namely,

$$\begin{aligned} \left[ \begin{array}{ll} \mathbf{M}_n + \frac{\Delta t_n}{2}\mathbf{C}_n +\frac{\Delta t_n^2}{4}\mathbf{K}_n &{} \mathbf{A}^\mathrm {T}_n\\ \mathbf{A}_n &{} \mathbf {0} \end{array}\right] \left[ \begin{array}{c} \mathbf{U}_{v,n}\\ \varvec{\lambda }^{*}_n \end{array}\right] = \left[ \begin{array}{c} \mathbf{F}_{v,n}-\frac{\Delta t_n}{2}\mathbf{F}_{u,n} \\ \mathbf {0} \end{array}\right] , \end{aligned}$$

with $\varvec{\lambda }^*_n :=\varvec{\lambda }_{v,n} - \frac{\Delta t_n}{2}\varvec{\lambda }_{u,n}$. Once the velocities $\mathbf{U}_{v,n}$ are known, the displacements are obtained solving the system

$$\begin{aligned} \left[ \begin{array}{ll} \mathbf{K}_n &{} \mathbf{A}^\mathrm {T}_n\\ \mathbf{A}_n &{} \mathbf {0} \end{array}\right] \left[ \begin{array}{ll} &{}\mathbf{U}_{u,n} \\ &{} \varvec{\lambda }_{u,n} \end{array}\right] = \left[ \begin{array}{l} \mathbf{F}_{u,n} +\frac{\Delta t_n}{2}\mathbf{K}_n\mathbf{U}_{v,n} \\ \mathbf {0} \end{array}\right] . \end{aligned}$$

(37)