Nonsmooth modal analysis of an elastic bar subject to a unilateral contact constraint

Abstract

This contribution proposes a numerical procedure capable of performing nonsmooth modal analysis (mode shapes and corresponding frequencies) of the autonomous wave equation defined on a finite one-dimensional domain with one end subject to a Dirichlet condition and the other end subject to a frictionless time-independent unilateral contact condition. Nonsmooth modes of vibration are defined as one-parameter continuous families of nonsmooth periodic orbits satisfying the local equation together with the linear and nonlinear boundary conditions. The analysis is performed using the wave finite element method, which is a shock-capturing finite volume method. As opposed to the traditional finite element method with time-stepping schemes, potentially discontinuous deformation, stress and velocity wave fronts induced by the unilateral contact condition are here accurately described, which is critical for seeking periodic orbits. Additionally, the proposed strategy introduces neither numerical dispersion nor artificial dissipation of energy, as required for modal analysis. As a consequence of the mixed time–space discretization, no impact law is needed for the well-posedness of the problem in line with the continuous framework. The frequency–energy dependency nonlinear spectrum of vibration, shown in the form of backbone curves, provides valuable insight on the dynamics. In contrast to the linear system (without the unilateral contact condition) whose modes of vibration are standing harmonic waves, the nonsmooth modes of vibrations are traveling waves stemming from the unilateral contact condition. It is also shown that the vibratory resonances of the periodically driven system with light structural damping are well predicted by nonsmooth modal analysis. Furthermore, the initially unstressed and prestressed configurations exhibit stiffening and softening behaviors, respectively, as expected.

Appendices

Appendix A: Description of the wave finite element method

In this section, the WFEM, introduced by Shorr for the simulation of shock wave propagation in solids [26], is thoroughly described.

1.1 A.1 Hyperbolic system of conservation laws

Local Eq. (1) can be equivalently written as a system of two first-order partial differential equations in terms of velocities v(x, t) and stresses $\sigma (x,t)$

$$\begin{aligned} \left. \begin{aligned} \sigma _{,t} - E v_{,x}&= 0 \\ \rho v_{,t} - \sigma _{,x}&= 0 \end{aligned} \;\right\} , \quad \forall x\in \mathopen {]}0\,;L\mathclose {[},\quad \forall t > 0 \end{aligned}$$

(19)

where $(\bullet )_{,t}$ is the derivation in time and $(\bullet )_{,x}$ is the derivation in space of quantity $(\bullet )$ [33]. Recall that axial strains $\varepsilon =u_{,x}$ are bounded by nonphysical inter-penetration, which translates into $u_{,x}> -\,1$, see Sect. 2. Displacements can be straightforwardly recovered by space integration of the strains $u(x,\cdot \,)=\int _0^x u_{,s}(s,\cdot \,)\mathop {}\!\mathrm {d}s - u(0,\cdot \,)$, where $u(0,\cdot \,)=0$. By posing $\mathbf {q}=[\sigma v]^\top$, Eq. (19) can be recast as

$$\begin{aligned} \mathbf {q}_{,t} + \mathbf {B} \mathbf {q}_{,x}=\mathbf {0}\quad \text {where}\quad \mathbf {B}=\begin{bmatrix} 0&-E \\ - 1/\rho&0 \end{bmatrix}. \end{aligned}$$

(20)

The eigenvalues of matrix $\mathbf {B}$ are $\lambda _1 =-\sqrt{E/\rho }$ and $\lambda _2 = \sqrt{E/\rho }$, coinciding with the algebraic propagation velocity of the elastic wave: positive and negative for the two waves propagating in opposite directions. Since both eigenvalues are distinct and real, Eq. (20) is also referred to as a hyperbolic system of conservation laws [33].

Equation (20) involves time and space derivatives of $\mathbf {q}$. However, observing that $\mathbf {q}_{,t} + \mathbf {B} \mathbf {q}_{,x}=\mathbf {0}$ is a local form of the conservation law of $\mathbf {q}$ (implying $\mathbf {q}(\,\cdot \,,t)$ can only change due to fluxes at the boundaries) corresponding to the following integral form

$$\begin{aligned} \frac{\mathop {}\!\mathrm {d}}{\mathop {}\!\mathrm {d}t}\Bigl (\int ^{x_2}_{x_1}\mathbf {q}(x,t) \mathop {}\!\mathrm {d}x\Bigr )= \mathbf {B}\bigl (\mathbf {q}(x_1,t)-\mathbf {q}(x_2,t)\bigr ), \end{aligned}$$

(21)

it appears that the condition on the smoothness of $\mathbf {q}$ is no longer required. Therefore, $\mathbf {q}$ is allowed to exhibit discontinuities in time and space [34].

1.2 A.2 Discretization

The WFEM consists in dividing the spatial and temporal domain into grid cells of equal size and keeping track of an approximation to the integral of $\mathbf {q}$ within every single cell. As depicted in Fig. 26, the bar is discretized using a uniform grid of N cells. Each ith cell, denoted by $\mathcal {C}_i$, is delimited by the interval $(x_{i-1/2},x_{i+1/2})$. Similarly, time is discretized into intervals of equal length $\varDelta t=t_{n+1}- t_n$. The average of $\mathbf {q}(\,\cdot \,,t)$ over the ith cell at time $t_n$ is

$$\begin{aligned} \mathbf {Q}_i^{(n)} = \frac{1}{\varDelta x} \int _{x_{i-1/2}}^{x_{i+1/2}} \mathbf {q}(x,t_{n})\mathop {}\!\mathrm {d}x = \frac{1}{\varDelta x} \int _{\mathcal {C}_i} \mathbf {q}(x,t_n)\mathop {}\!\mathrm {d}x \end{aligned}$$

(22)

where $\varDelta x = x_{i+1/2} - x_{i-1/2}$ is the length of cell i. The integral form of conservation law (21) applied to cell $\mathcal {C}_i$ reads

$$\begin{aligned} \frac{\mathop {}\!\mathrm {d}}{\mathop {}\!\mathrm {d}t}\int _{\mathcal {C}_i}\mathbf {q}(x,t)\mathop {}\!\mathrm {d}x=\mathbf {B}\bigl (\mathbf {q}(x_{i-1/2},t) - \mathbf {q}(x_{i+1/2},t)\bigr ). \end{aligned}$$

(23)

This expression is now employed to develop an explicit time-stepping algorithm where $\mathbf {Q}_i^{(n+1)}$ is approximated by a function of $\mathbf {Q}_i^{(n)}$. Equation (23) is integrated between $t_n$ and $t_{n+1}$ yielding

$$\begin{aligned} \begin{aligned}&\int _{\mathcal {C}_i}\bigl (\mathbf {q}(x,t_{n+1})-\mathbf {q}(x,t_{n})\bigr )\mathop {}\!\mathrm {d}x \\&\quad =\int _{t_n}^{t_{n+1}} \bigl (\mathbf {B}\bigl (\mathbf {q}(x_{i-1/2},t) - \mathbf {q}(x_{i+1/2},t)\bigr )\bigr )\mathop {}\!\mathrm {d}t. \end{aligned} \end{aligned}$$

(24)

Fig. 26

Discretization of the spatial domain in grid cells at time $t_n$

Rearranging and dividing by $\varDelta x$ leads to

$$\begin{aligned} \frac{1}{\varDelta x}\int _{\mathcal {C}_i}\mathbf {q}(x,t_{n+1})\mathop {}\!\mathrm {d}x= & {} \frac{1}{\varDelta x}\Bigl ( \int _{\mathcal {C}_i}\mathbf {q}(x,t_{n})\mathop {}\!\mathrm {d}x \nonumber \\&-\int _{t_n}^{t_{n+1}} \mathbf {B}\mathbf {q}(x_{i+1/2},t)\mathop {}\!\mathrm {d}t \nonumber \\&+\int _{t_n}^{t_{n+1}} \mathbf {B}\mathbf {q}(x_{i-1/2},t)\mathop {}\!\mathrm {d}t\Bigr ).\nonumber \\ \end{aligned}$$

(25)

This equation describes how the cell average should be updated within a time step in order to satisfy the conservation of $\mathbf {q}$. In general, the two integrals involving $\mathbf {B}\mathbf {q}$ on the right-hand side of the equation cannot be evaluated exactly. Following [35], we pose

$$\begin{aligned} \mathbf {F}_{i\pm 1/2}^{(n)} \approx \frac{1}{\varDelta t}\int _{t_n}^{t_{n+1}}\mathbf {B}\mathbf {q}(x_{i\pm 1/2},t)\mathop {}\!\mathrm {d}t \end{aligned}$$

(26)

and Eq. (25) simply becomes

$$\begin{aligned} \mathbf {Q}^{(n+1)}_i=\mathbf {Q}^{(n)}_i-\frac{\varDelta t}{\varDelta x}\bigl (\mathbf {F}_{i+1/2}^{(n)}-\mathbf {F}_{i-1/2}^{(n)}\bigr ). \end{aligned}$$

(27)

The next subsection is dedicated to the computation of $\mathbf {F}_{i\pm 1/2}^{(n)}$, which are the time-averaged fluxes at $x=x_{i\pm 1/2}$.

Fig. 27

Reconstruction of $\mathbf {q}(x,t_n)$ from the average fluxes $\mathbf {Q}^{(n)}_i$

1.3 A.3 Approximation of the time–averaged fluxes

To approximate the fluxes at the interfaces defined by Eq. (26), the state $\mathbf {q}(x,t_n)$ at time $t_n$ is assumed to be a piecewise constant function defined for all x, constructed from the cell averages $\mathbf {Q}^{(n)}_i$ as depicted in Fig. 27. This piecewise reconstruction of the function $\mathbf {q}(x,t_n)$ is identical to the Godunov’s approach widely employed in computational fluid dynamics [36]. A suitable approximation of the flux $\mathbf {F}_{i+1/2}^{(n)}$ can be obtained by solving the problem, either numerically or exactly, of conservation law Eq. (20) together with the following discontinuous conditions at time $t_n$ [36]:

$$\begin{aligned} \begin{aligned} \sigma (x,t_n)= {\left\{ \begin{array}{ll} ~\sigma ^{(n)}_{i} &{} \text {if }x\leqslant x_{i+1/2}\\ ~\sigma ^{(n)}_{i+1} &{} \text {if }x>x_{i+1/2} \end{array}\right. }\\ v(x,t_n)= {\left\{ \begin{array}{ll} ~v^{(n)}_{i} &{} \text {if }x\leqslant x_{i+1/2}\\ ~v^{(n)}_{i+1} &{} \text {if }x>x_{i+1/2} \end{array}\right. } \end{aligned} \end{aligned}$$

(28)

which constitutes a Riemann problem centered at $x_{i+1/2}$ between cells $\mathcal {C}_i$ and $\mathcal {C}_{i+1}$ [36]. The solution to this Riemann problem consists of two shock waves propagating along the characteristic lines $x=\pm ct$, one moving to the left into cell $\mathcal {C}_i$ and one moving to the right into cell $\mathcal {C}_{i+1}$ as depicted in a space–time plot in Fig. 28. The shock wave traveling to the left, indicated by $\mathcal {W}_1$, propagates at velocity $s_1$ and connects the state $\mathbf {Q}^{(n)}_i$ and the interior state $\mathbf {Q}_i^*$ generated by such shock wave. Moreover, the solution $t\mapsto \mathbf {q}(x_{i+1/2},t)$ is constant over the time interval $[t_n,t_{n+1}]$.

Fig. 28

Structure of the solution to the Riemann problem depicted in a space–time plot

The Rankine–Hugoniot jump condition is proven to hold across any propagating discontinuity [37] which can be written for the left wave $\mathcal {W}_1$ propagating at velocity $s_1$ and the right wave $\mathcal {W}_2$ propagating at velocity $s_2$:

$$\begin{aligned} \begin{aligned} \mathcal {W}_1&\rightarrow {\left\{ \begin{array}{ll} ~ s_1 \bigl (\sigma ^*_{i} - \sigma ^{(n)}_{i}\bigr ) = - E\bigl (v^*_{i} - v^{(n)}_{i}\bigr ), \\ ~ \rho s_1 \bigl (v^*_{i} - v^{(n)}_{i}\bigr ) = -\bigl (\sigma ^*_{i} - \sigma ^{(n)}_{i}\bigr ) \end{array}\right. }\\ \mathcal {W}_2&\rightarrow {\left\{ \begin{array}{ll} ~ s_2\bigl (\sigma ^{(n)}_{i+1}-\sigma ^*_{i+1}\bigr )=-E\bigl (v^{(n)}_{i+1} - v^*_{i+1}\bigr ),\\ ~ \rho s_2 \bigl (v^{(n)}_{i+1} - v^*_{i+1}\bigr ) = -\bigl (\sigma ^{(n)}_{i+1} - \sigma ^*_{i+1}\bigr ). \end{array}\right. } \end{aligned} \end{aligned}$$

(29)

Because of material continuity, cells $\mathcal {C}_i$ and $\mathcal {C}_{i+1}$ cannot separate. This requires that the interior states must be equal across the material interface, $\mathbf {Q}^*_{i+1/2}=\mathbf {Q}_{i+1}^*=\mathbf {Q}_{i}^*$. By knowing that the shock speeds $s_1 = - s_2 = -\sqrt{E/\rho }=-c$ are known and constants, the intermediate state is approximated with $\mathbf {q}(x_{i+1/2},t)\approx \mathbf {Q}_{i+1/2}^*$ such that $t_{n}\leqslant t\leqslant t_{n+1}$ and can be calculated from Eq. (29):

$$\begin{aligned} \mathbf {Q}^*_{i+1/2}= & {} \begin{bmatrix} \sigma _{i+1/2}^* \\ v_{i+1/2}^* \end{bmatrix}\nonumber \\= & {} \frac{1}{2}\!\begin{bmatrix}\! \sigma ^{(n)} _{i+1}\!+\! \sigma ^{(n)} _{i} + \rho c \bigl (v^{(n)}_{i+1}-v^{(n)}_{i}\bigr )\\ v^{(n)} _{i+1} \!+\! v^{(n)} _{i}\!+\! \frac{1}{\rho c} \bigl (\sigma ^{(n)}_{i+1}-\sigma ^{(n)}_{i}\bigr ) \!\end{bmatrix}\!\!. \end{aligned}$$

(30)

Equation (30) is regarded as the exact solution of the Riemann problem involving linear elastodynamics [35, 38]. The flux approximation in Eq. (26) can be calculated with the solution of a Riemann problem at the cell interface as

$$\begin{aligned} \mathbf {F}_{i+1/2}^{(n)} \approx \frac{\mathbf {B}}{\varDelta t} \int _{t_n}^{t_{n+1}}\mathbf {Q}^*_{i+1/2}\mathop {}\!\mathrm {d}t \approx \mathbf {B} \mathbf {Q}^*_{i+1/2}. \end{aligned}$$

(31)

In a nonlinear framework for the local equation, a solution to the Riemann problem should be approximated numerically using Riemann solvers [39].

1.4 A. 4 Formulation for inner grid cells

Inserting Eq. (31) into Eq. (27) produces the iterative scheme

$$\begin{aligned} \mathbf {Q}^{(n+1)}_i = \mathbf {Q}^{(n)}_i - \frac{\varDelta t}{\varDelta x}\bigl (\mathbf {B} \mathbf {Q}^*_{i+1/2} - \mathbf {B} \mathbf {Q}^*_{i-1/2}\bigr ). \end{aligned}$$

(32)

Equation (32) describes the evolution in time of the states of the grid cells $\mathcal {C}_i$. This subsection provides the formulation for the inner cells, where $i=2,\ldots ,N-1$. The boundary cells $\mathcal {C}_1$ and $\mathcal {C}_N$ require a different treatment explained in the next subsection. Expressing the flux approximation, employing Eq. (31) on the right side of an inner cell yields

$$\begin{aligned} \mathbf {B} \mathbf {Q}^*_{i+1/2}{=}\frac{1}{2} \begin{bmatrix} -E\bigl (v^{(n)}_{i+1}+v^{(n)}_{i}+\frac{1}{\rho c}\bigl (\sigma ^{(n)}_{i+1}-\sigma ^{(n)}_{i}\bigr )\bigr )\\ -\frac{1}{\rho }\bigl (\sigma ^{(n)} _{i+1} + \sigma ^{(n)} _{i}+\rho c\bigl (v^{(n)}_{i+1}-v^{(n)}_{i}\bigr )\bigr )\end{bmatrix}. \end{aligned}$$

(33)

Performing the same operation on the left side of the cell reads

$$\begin{aligned} \mathbf {B} \mathbf {Q}^*_{i-1/2} {=} \frac{1}{2} \begin{bmatrix} -E\bigl ( v^{(n)} _{i} + v^{(n)}_{i-1} + \frac{1}{\rho c}\bigl (\sigma ^{(n)}_{i}-\sigma ^{(n)}_{i-1}\bigr ) \bigr ) \\ -\frac{1}{\rho }\bigl (\sigma ^{(n)} _{i} + \sigma ^{(n)} _{i-1} + \rho c \bigl (v^{(n)}_{i}-v^{(n)}_{i-1}\bigr )\bigr ) \end{bmatrix}. \end{aligned}$$

(34)

Accordingly, the total flux within an inner cell is the quantity $\mathbf {B} \mathbf {Q}^*_{i+1/2} - \mathbf {B} \mathbf {Q}^*_{i-1/2}$ which when substituted into Eq. (32) yields

$$\begin{aligned}&\begin{bmatrix} \sigma ^{(n+1)} _i\\v^{(n+1)}_i \end{bmatrix}\nonumber \\&\quad =\begin{bmatrix} \sigma ^{(n)} _i \\ v^{(n)}_i \end{bmatrix}+\frac{\varDelta t}{2 \varDelta x} \nonumber \\&\quad \times \begin{bmatrix} E\bigl (v^{(n)}_{i+1} - v^{(n)}_{i-1}\bigr ) + c\bigl (\sigma ^{(n)}_{i+1}-2\sigma ^{(n)}_{i} + \sigma ^{(n)}_{i-1}\bigr )\\ \frac{1}{\rho c} \bigl (\sigma ^{(n)}_{i+1} - \sigma ^{(n)}_{i-1}\bigr ) + c \bigl (v^{(n)}_{i+1}-2v^{(n)}_{i}+v^{(n)}_{i-1}\bigr ) \end{bmatrix}.\nonumber \\ \end{aligned}$$

(35)

Since the exact solution of a Riemann problem is being used, WFEM incorporates an appropriate time step $\varDelta t=\varDelta x/c$. Then, the stress and velocity of a grid cell $\mathcal {C}_i$ at time $t_{n+1}$ are calculated as

$$\begin{aligned} \begin{bmatrix} \sigma ^{(n+1)}_i \\ v^{(n+1)}_i \end{bmatrix} = \frac{1}{2} \begin{bmatrix} \sigma ^{(n)}_{i+1} + \sigma ^{(n)}_{i-1} + \rho c \bigl (v^{(n)}_{i+1} - v^{(n)}_{i-1}\bigr ) \\ v^{(n)}_{i+1}+v^{(n)}_{i-1} + \frac{1}{\rho c}\bigl (\sigma ^{(n)}_{i+1} - \sigma ^{(n)}_{i-1}\bigr ) \end{bmatrix}. \end{aligned}$$

(36)

The above strong assumption is suitable only for 1D elastodynamics problems, since the wave velocity and the direction of the propagation is known. Also, such assumption enforces energy conservation and eliminates numerical dissipation [26]. In the multidimensional framework, even though the waves velocities are known, the waves could propagate in various directions throughout the physical domain.

Equation (36) provides the main equation of the WFEM and characterizes how the average value $\mathbf {Q}_i^{(n)}$ of $\mathbf {q}$ in an inner cell $\mathcal {C}_i$ is updated at each time step. As required by the local conservation law (21) resulting from the absence of body forces, the evolution of the state of the inner cells depends only on the values of the adjacent cells. WFEM can be seen as the transference of the whole information embedded in cell $\mathcal {C}_i$ to its adjacent cells at each time step. Employing the latter approach to obtain the evolution of cell states, involving discontinuities such as shock and rarefaction waves, is well known by the Fluid Mechanics community employing finite volume methods [36].

1.5 A.5 Formulation for boundary grid cells

To compute the state of the boundary grid cells, the computational domain is extended by including additional cells on both boundaries, known as ghost cells [35], whose average values depend on the boundary conditions. This concept is taken from the finite volume methods. Figure 29 depicts ghost cells for a system discretized using N cells.

Fig. 29

Computational space domain with ghost cells

Equation (36) can then be used to update the average value on the boundary cells, which have now become inner cells. Only a single ghost cell is required at each boundary because the computation of the average value depends only on the states of the adjacent cells. For instance, the fixed–free elastic bar without the complementarity conditions of contact satisfies the two boundary conditions $u(0,t)=v(0,t)=0$ and $E u_{,x}(L,t)=\sigma (L,t)=0$. These conditions are used to define the average values within the ghost cells as follows:

$$\begin{aligned} \begin{aligned}&\text {At ghost cell}\,\, \mathcal {C}_0:\quad \left\{ \begin{aligned} \,&\sigma _0=\sigma _1 \\&v_0=-v_1, \end{aligned}\right. \quad \\&\text {At ghost cell }\mathcal {C}_{N+1}:\quad \left\{ \begin{aligned} \,&\sigma _{N+1}=-\sigma _N \\&v_{N+1}=v_N. \end{aligned}\right. \end{aligned} \end{aligned}$$

(37)

Such average values coincide with the theory of reflection of elastic waves from fixed and free boundaries [2], which states that stress waves reflect from a fixed boundary with the same sign and from a free boundary with the changed sign; similarly, velocity waves reflect from a fixed boundary with an opposite sign and from a free boundary with the same sign. The evolution of the average values of the boundary cells, $\mathcal {C}_1$ and $\mathcal {C}_N$, can be calculated by introducing the average values of Eq. (37) into Eq. (36), yielding for $\mathcal {C}_1$:

$$\begin{aligned} \mathbf {Q}_1^{(n+1)} = \frac{1}{2} \begin{bmatrix} \sigma ^{(n)}_{2} + \sigma ^{(n)}_{1} + \rho c \bigl (v^{(n)}_{2} + v^{(n)}_{1}\bigr ) \\ v^{(n)}_{2}-v^{(n)}_{1} + \frac{1}{\rho c}\bigl (\sigma ^{(n)}_{2} - \sigma ^{(n)}_{1}\bigr ) \end{bmatrix} \end{aligned}$$

(38)

and for $\mathcal {C}_{N}$:

$$\begin{aligned} \mathbf {Q}_N^{(n+1)} = \frac{1}{2} \begin{bmatrix} \sigma ^{(n)}_{N-1} -\sigma ^{(n)}_{N} + \rho c \bigl (v^{(n)}_{N} + v^{(n)}_{N-1}\bigr ) \\ v^{(n)}_{N}+v^{(n)}_{N-1} - \frac{1}{\rho c}\bigl (\sigma ^{(n)}_{N} + \sigma ^{(n)}_{N-1}\bigr ) \end{bmatrix}. \end{aligned}$$

(39)

1.6 A.6 WFEM generic algorithm

Altogether, the previous derivations lead to a set of equations that describe how $\mathbf {q}$ is updated in time for every cell. The different steps are summarized in Algorithm 1. Using $\varDelta t= \varDelta x/ c$ it computes, in the framework of linear elastodynamics, the propagation at finite speed c of a wave and accounts for the reflection conditions at the boundaries. By definition of $\varDelta t$, the CFL condition $\varDelta t\le \varDelta x/c$ is always satisfied, so the method is always stable [35, 40]. Additionally, because the global error is proportional to the discretization steps, the WFEM is first-order accurate in both space and time [26].

1.7 A.7 Matrix formulation

Similar to other numerical methods applied on linear systems, the WFEM can be rewritten in a convenient matrix form which facilitates the process of finding nonsmooth modes of vibration. More specifically, the state vector of the system $\mathbf {Q}^{(n)}=\bigl [\mathbf {Q}_1^{(n)} \ldots \mathbf {Q}_N^{(n)}\bigr ]^{\top }\in \mathbb {R}^{2N}$ at time $t_n$ satisfies the identity²

$$\begin{aligned} \mathbf {Q}^{(n)}=\mathbf {A}\mathbf {Q}^{(n-1)},\quad \forall n\ge 1 \end{aligned}$$

(40)

and $\mathbf {Q}^{(0)}$ is the initial state. The matrix $\mathbf {A}$ gathers stiffness and inertial terms as well as the type of boundary conditions. It is now derived for the fixed–free BC. In a matrix form, Eq. (36), which governs the evolution of inner cells, reads

$$\begin{aligned} \begin{bmatrix} \sigma ^{(n+1)}_i \\ v^{(n+1)}_i \end{bmatrix} = \frac{1}{2} \begin{bmatrix} 1&0&1&-\rho c&0&\rho c\\ -\frac{1}{\rho c}&0&\frac{1}{\rho c}&1&0&1 \end{bmatrix} \begin{bmatrix} \sigma ^{(n)}_{i-1} \\ \sigma ^{(n)}_{i} \\ \sigma ^{(n)}_{i+1} \\ v^{(n)}_{i-1} \\ v^{(n)}_{i} \\ v^{(n)}_{i+1} \end{bmatrix}. \end{aligned}$$

(41)

For the boundary cells in Eqs. (38) and (39), the matrix form follows as

$$\begin{aligned} \begin{bmatrix} \sigma ^{(n+1)}_1 \\ v^{(n+1)}_1 \end{bmatrix} = \frac{1}{2} \begin{bmatrix} 1&1&\rho c&\rho c\\ -\frac{1}{\rho c}&\frac{1}{\rho c}&-1&1 \end{bmatrix} \begin{bmatrix} \sigma ^{(n)}_{1} \\ \sigma ^{(n)}_{2} \\ v^{(n)}_{1} \\ v^{(n)}_{2} \end{bmatrix} \end{aligned}$$

(42)

and

$$\begin{aligned} \begin{bmatrix} \sigma ^{(n+1)}_N \\ v^{(n+1)}_N \end{bmatrix} = \frac{1}{2} \begin{bmatrix} 1&-1&\rho c&\rho c\\ -\frac{1}{\rho c}&-\frac{1}{\rho c}&1&1 \end{bmatrix} \begin{bmatrix} \sigma ^{(n)}_{N-1} \\ \sigma ^{(n)}_{N} \\ v^{(n)}_{N-1} \\ v^{(n)}_{N} \end{bmatrix}. \end{aligned}$$

(43)

Then, the block matrix $\mathbf {A}\in \mathbb {R}^{2N\times 2N}$ can be constructed using four $N\times N$ matrices $\mathbf {A}_1,\mathbf {A}_2,\mathbf {A}_3,\mathbf {A}_4$ whose expression can be derived from

$$\begin{aligned} \mathbf {A}_{\text {G}}(a,b,c,d) =\frac{1}{2} \begin{bmatrix} a&b&0&\dots&0&0&0\\ c&0&b&0&0&0\\ 0&c&0&0&0&0\\ \vdots&\ddots&\ddots&\ddots&\vdots \\ 0&0&0&0&b&0\\ 0&0&0&c&0&b\\ 0&0&0&\dots&0&c&d\\ \end{bmatrix} \end{aligned}$$

(44)

with $\mathbf A_1{=}\mathbf {A}_{\text {G}}(-1,1,1,1)$, $\mathbf A_2{=} \rho c\mathbf {A}_{\text {G}}(-1,1,-1,-1)$, $\mathbf A_3= \mathbf {A}_{\text {G}}(1,1,-1,1)/\rho c$, and $\mathbf A_4= \mathbf {A}_{\text {G}}(1,1,1,-1)$. Then, block matrix $\mathbf {A}$ is written as

$$\begin{aligned} \mathbf A = \begin{bmatrix} \mathbf A_1&\mathbf A_2 \\ \mathbf A_3&\mathbf A_4 \end{bmatrix}. \end{aligned}$$

(45)

Another matrix $\mathbf {A}$ can be constructed in the same way for the fixed–fixed BC. Finally, the unknown $\mathbf {Q}^{(n)}$ can be directly expressed in terms of the initial conditions $\mathbf {Q}^{(0)}$ from Eq. (40) by

$$\begin{aligned} \mathbf {Q}^{(n)}=\mathbf {A}^{n}\mathbf {Q}^{(0)} \end{aligned}$$

(46)

where $\mathbf {A}^{n}$ is known for each type of BC.

Fig. 30

Conditional switch for contact treatment in WFEM. a Inactive contact. b Active contact

Appendix B: Treatment of unilateral contact in WFEM

The unilateral contact constraints involved in the formulation are enforced using the concept of floating boundary conditions [26] which can be regarded as a conditional switch between fixed–free and fixed–fixed boundary conditions [41] when a penetration is detected during a time iteration, as illustrated in Fig. 30. In the continuous framework, these two boundary conditions are

• Fixed–free BC (inactive contact)

  $$\begin{aligned} u(0,t)=v(0,t)=0,\quad E u_{,x}(L,t)=\sigma (L,t)=0 \end{aligned}$$

  (47)

• Fixed–fixed BC (active contact)

  $$\begin{aligned} u(0,t)=v(0,t)=0,\quad u(L,t)=g_0 \rightarrow v(L,t)=0. \end{aligned}$$

  (48)

The gap function g(u(L, t)), which is a function of the displacement u(L, t), but not an explicit function of t, is discretized in time to calculate a possible penetration of the bar as

$$\begin{aligned} g^{(n)}=g_0 - u_{N+1/2}^{(n)} \end{aligned}$$

(49)

where $u_{N+1/2}^{(n)}$ is the displacement of the contacting interface which can be calculated by numerically integrating strains of the bar $u^{(n)}_{,x}$ at time $t_n$. This equality is used at each time step to check whether contact is active during the next iteration: if $g^{(n)}>0$, a free boundary condition is enforced while if $g^{(n)}\leqslant 0$, a fixed boundary condition is considered via the change of matrix $\mathbf {A}$.

Based on the theory of reflection of elastic waves from boundaries [2], the state of the ghost cell $\mathcal {C}_{N+1}$ is updated as follows

• Active contact [$g^{(n)}\leqslant 0$]

  $$\begin{aligned} \left\{ \begin{aligned} \,&\sigma _{N+1}=\sigma _N \\&v_{N+1}=-v_N \\ \end{aligned}\right. \end{aligned}$$

  (50)

• Inactive contact [$g^{(n)}>0$]

  $$\begin{aligned} \left\{ \begin{aligned} \,&\sigma _{N+1}=-\sigma _N \\&v_{N+1}=v_N \\ \end{aligned}\right. \end{aligned}$$

  (51)

Equation (36) is subsequently used to calculate the evolution of the average values inside the boundary cell $\mathcal {C}_N$, as detailed in Sect. 1. Additionally, the contact force $r^{(n)}$ is calculated by employing Eq. (30):

$$\begin{aligned} r^{(n)} = \mathcal {S}\bigl (\sigma ^{(n)}_N - \rho c v^{(n)}_N\bigr ). \end{aligned}$$

(52)

The sign of this quantity is tracked to locate the time of release, when the bar returns to free condition at $x=L$. Algorithm 1 is modified to include the floating boundary conditions as described in Algorithm 2.

From the matrix formulation of the WFEM in Eq. (40), two matrices $\mathbf {A}$ shall then be distinguished: $\mathbf {A}_{\text {f}}$ for the fixed–free condition (no contact) and $\mathbf {A}_{\text {c}}$ for the fixed–fixed condition (contact). Both matrices embed the same stiffness and inertial terms of the system of interest; the only unshared information are the boundary conditions. To summarize, the developed WFEM with floating boundary conditions is a numerically conservative and stable scheme able to properly propagate shock waves induced by a switch in the boundary conditions, the latter being governed by complementarity constraints.