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A dynamic hybrid local/nonlocal continuum model for wave propagation

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Abstract

In this work, we develop a dynamic hybrid local/nonlocal continuum model to study wave propagations in a linear elastic solid. The developed hybrid model couples, in the dynamic regime, a classical continuum mechanics model with a bond-based peridynamic model using the Morphing coupling method that introduced in a previous study (Lubineau et al., J Mech Phys Solids 60(6):1088–1102, 2012). The classical continuum mechanical model is known as a local continuum model, while the peridynamic model is known as a nonlocal continuum model. This dynamic hybrid model aims to introduce the nonlocal model into the key structural domain, in which the dispersions or crack nucleations may occur due to flaws, while applying the local model to the rest of the structural domain. Both the local and nonlocal continuum domains are overlapped in the coupled subdomain. We study the speeds and angular frequencies of the plane waves, with small and large wavenumbers obtained by the hybrid model and compare them to purely local and purely nonlocal solutions. The error of the hybrid model is discussed by analyzing the ghost forces, and the work done by the ghost forces is considered equivalent to the energy of spurious reflections. One- and two-dimensional numerical examples illustrate the validity and accuracy of the proposed approach. We show that this dynamic hybrid local/nonlocal continuum model can be successfully applied to simulate wave propagations and crack nucleations induced by waves.

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Acknowledgements

The authors gratefully acknowledge the financial support received from the National Natural Science Foundation (11872016), the Fundamental Research Funds of Dalian University of Technology (Grant No. DUT20RC(5)005), and KAUST baseline funding for the completion of this work. Helpful discussions with Dr. Youwei Zhang from Dalian University of Technology and with Dr. Meizhen Xiang from Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics are also gratefully acknowledged.

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Appendix A: Numerical discretization and algorithm procedure

Appendix A: Numerical discretization and algorithm procedure

The numerical discretization and algorithm procedure for the hybrid equations of motion (i.e., Eqs. (6)–(13)) is stated below. We consider the spatial and temporal variables separately. The equations are first discretized on the spatial dimension by the finite element method. The approach is the same as for the static analysis. Then, the resulting equation, which involves the second derivative of displacement with respect to time, is discretized on the temporal dimension by the finite difference method. Finally, we list the algorithm procedures to solve the hybrid equation of motion.

1.1 Appendix A.1: The spatial discretization by finite element method

We divide the whole domain, \(\Omega \), by a finite number of elements, \(V_i, i=1,2,\ldots , n\), where n is the number of elements. These elements are nonoverlapping, but common vertices, called “mesh nodes”, are shared between adjacent elements. Thus, we can write that \(\Omega = V_1 \cup V_2 \cup \cdots \cup V_n\). Because it is defined in Sect. 3 that \(\Omega = \Omega _1 \cup \Omega _m \cup \Omega _2\), \(\Omega _{1} \cap \Omega _{2} = \emptyset \), \(\Omega _{1} \cap \Omega _{m} = \emptyset \) and \(\Omega _{2} \cap \Omega _{m} = \emptyset \), we assume for definiteness and without a loss of generality that \(\Omega _1 = V_1 \cup V_2 \cup \cdots \cup V_{n'}\), \(\Omega _m = V_{n'+1} \cup V_{n'+2} \cup \cdots \cup V_{n''}\) and \(\Omega _2 = V_{n''+1} \cup V_{n''+2} \cup \cdots \cup V_{n}\), for \(1<n'<n''<n\). In addition, we define the divisions of the neighborhood of any point \(\varvec{x}\) (i.e., \(\mathcal {H}_{\delta }(\varvec{x})\cap \Omega \)). Indeed, there exists a minimal set of elements, \(\mathcal {A}_{\varvec{x}}\), which is defined by \(\mathcal {A}_{\varvec{x}}=\{V_{\varvec{x}}^1, V_{\varvec{x}}^2, \ldots , V_{\varvec{x}}^{h({\varvec{x}})}\} \subset \{V_1, V_2, \ldots , V_n\}\), such that \((\mathcal {H}_{\delta }(\varvec{x})\cap \Omega ) \subset \mathcal {B}_{\varvec{x}}, \; \forall \varvec{x}\in \Omega \backslash \Omega _1\), where \(\mathcal {B}_{\varvec{x}}=V_{\varvec{x}}^{1} \cup V_{\varvec{x}}^{2} \cdots \cup V_{\varvec{x}}^{h({\varvec{x}})}\) and \(h({\varvec{x}})\) denotes the number of elements in \(\mathcal {A}_{\varvec{x}}\). By reassigning a value of 0 to \(c^0 (\left| {\varvec{\xi }}\right| )\), where \(\varvec{\xi }=\varvec{x}'-\varvec{x}\), for \(\varvec{x}' \in \mathcal {B}_{\varvec{x}}\backslash \mathcal {H}_{\delta }(\varvec{x})\), the coefficient function of micromodulus \(c^0 (\left| {\varvec{\xi }}\right| )\) is extended to the whole domain \(\mathcal {B}_{\varvec{x}}\). Then, a virtual work equation corresponding to Eqs. (6)–(13) can be written by replacing the subdomain \((\mathcal {H}_{\delta }(\varvec{x})\cap \Omega )\) with the subdomain \(\mathcal {B}_{\varvec{x}}\) for integration. That is

$$\begin{aligned}&\int _{\Omega } P\left( \varvec{u}, t\right) d\Omega + \int _{\Omega } W_c\left( \varvec{\varepsilon }, t\right) d\Omega +\int _{\Omega } W_p\left( {\varvec{\eta }}, t\right) d\Omega \nonumber \\&= \int _{\Omega } Q_b \left( \varvec{u}, t \right) d\Omega + \int _{\Gamma _{\varvec{T}}} Q_T \left( \varvec{u}, t \right) dS, \end{aligned}$$
(A.1)

where

$$\begin{aligned} P\left( \varvec{u}, t\right)= & {} \varvec{u}(\varvec{x}, t) \cdot \left( \rho (\varvec{x})\ddot{\varvec{u}}(\varvec{x},t)\right) , \\ W_c \left( \varvec{\varepsilon }, t\right)= & {} \frac{1}{2}\varvec{\varepsilon }\left( {\varvec{x}} , t \right) : {\varvec{E}} \left( {\varvec{x}} \right) :\varvec{\varepsilon }\left( {\varvec{x}} , t \right) , \\ W_p\left( {\varvec{\eta }}, t \right)= & {} \frac{1}{4} \int _{\mathcal {B}_{\varvec{x}}} c^0 (\left| {\varvec{\xi }}\right| ) \frac{\alpha ( {\varvec{x}} )+\alpha ( {\varvec{x}'} )}{2} \left( \varvec{\xi }\cdot \varvec{\eta }( \varvec{x}', \varvec{x}, t)\right) ^2 dV_{\varvec{x}'}, \\ Q_b \left( \varvec{u}, t \right)= & {} \varvec{u}(\varvec{x}, t) \cdot {\varvec{b}}(\varvec{x}, t), \;\; \text {and}\\ Q_T \left( \varvec{u}, t \right)= & {} \varvec{u}(\varvec{x}, t) \cdot {\overline{\varvec{T}}}( {\varvec{x}}, t ). \end{aligned}$$

The displacement solution can be approximately expressed in the finite element scheme using piecewise interpolation techniques. Let \(\varvec{u}_i\) denote the displacement solution over the element \(V_i\), which is given by

$$\begin{aligned} \varvec{u}_i(\varvec{x}, t) = \varvec{N}_i(\varvec{x})\varvec{d}_i(t), \quad \forall i=1,2,\ldots , n, \end{aligned}$$
(A.2)

where \(\varvec{N}_i\) is the matrix of shape functions and \(\varvec{d}_i\) is the nodal displacement. Furthermore, let the number of mesh nodes in the global discretized domain be m, then all nodal displacements can be denoted as \(\varvec{d}^{\text {T}}=\{d_1, d_2, \ldots , d_m\}\). Thus, for the nodal displacement \(\varvec{d}_i\) in any element \(V_i\), we know that \(\varvec{d}_i \subset \varvec{d}, \;\; \forall i=1,2,\ldots ,n\). Moreover, one can define that

$$\begin{aligned} \varvec{d}_i(t) = \varvec{R}_i\varvec{d}(t), \quad \forall i=1,2,\ldots , n, \end{aligned}$$
(A.3)

where \(\varvec{R}_i\) is a diagonal matrix in which the diagonal entries may be 0 or 1, depending on the nodes of element \(V_i\). Substituting Eq. (A.3) into the Eq. (A.2), it yields

$$\begin{aligned} \varvec{u}_i(\varvec{x}, t) = \varvec{N}_i(\varvec{x})\varvec{R}_i\varvec{d}(t), \quad \forall i=1,2,\ldots , n. \end{aligned}$$
(A.4)

Then, we use the variational method to Eq. (A.1) and, by substituting Eq. (A.4), it can be recast as follows:

$$\begin{aligned} \varvec{M}\ddot{\varvec{d}}(t) + \varvec{K}{\varvec{d}}(t) = \varvec{F}(t), \end{aligned}$$
(A.5)

where \(\ddot{\varvec{d}}\) denotes the nodal acceleration, \(\varvec{M}\), \(\varvec{K}\), and \(\varvec{F}(t)\) are the mass matrix, stiffness matrix, and loading vector, respectively, which have the following expressions:

$$\begin{aligned} \varvec{M}= & {} \displaystyle {\sum _{i=1}^{n}} \int _{V_i} \rho (\varvec{x}) \left[ \varvec{N}_i(\varvec{x})\varvec{R}_i \right] ^{\text {T}}\left[ \varvec{N}_i(\varvec{x}) \varvec{R}_i \right] dV_{\varvec{x}}, \\ \varvec{K}= & {} \displaystyle {\sum _{i=1}^{n}} \int _{V_i} \left[ \varvec{H}\varvec{N}_i(\varvec{x})\varvec{R}_i \right] ^{\text {T}}\left[ \varvec{E}\left( \varvec{x}\right) \right] \left[ \varvec{H}\varvec{N}_i(\varvec{x}) \varvec{R}_i \right] dV_{\varvec{x}} \\&+ \frac{1}{2} \displaystyle {\sum _{i=1}^{n}}\displaystyle {\sum _{j=1}^{h(\varvec{x})}}\int _{V_i} \int _{V_{\varvec{x}}^j} c^0 (\left| {\varvec{\xi }}\right| ) \frac{\alpha ( {\varvec{x}} )+\alpha ( {\varvec{x}'} )}{2} \left[ \varvec{N}_j(\varvec{x}')\varvec{R}_j - \varvec{N}_i(\varvec{x})\varvec{R}_i \right] ^{\text {T}} \\&\left[ \varvec{\xi }\otimes \varvec{\xi }\right] \left[ \varvec{N}_j(\varvec{x}')\varvec{R}_j - \varvec{N}_i(\varvec{x})\varvec{R}_i \right] dV_{\varvec{x}'}dV_{\varvec{x}} \; \text {and}\\ \varvec{F}(t)= & {} \displaystyle {\sum _{i=1}^{n}} \int _{V_i} \left[ \varvec{N}_i(\varvec{x})\varvec{R}_i\right] ^{\text {T}} \{\varvec{b}(\varvec{x}, t)\} dV_{\varvec{x}} \\&+ \displaystyle {\sum _{i=1}^{n}} \int _{S_i} \left[ \varvec{N}_i(\varvec{x})\varvec{R}_i\right] ^{\text {T}} \{\overline{\varvec{T}}( {\varvec{x}}, t )\} dS_{\varvec{x}}. \end{aligned}$$

where \(\varvec{H}\) denotes a matrix of differential operators (see Eq (25)), the notations \(\left[ \cdot \right] \) and \(\{\cdot \}\) denote a matrix and a vector, respectively, and \(S_i\) is the boundary of \(V_i\).

1.2 Appendix A.2: The temporal discretization by finite difference method

Here, the central difference method is employed. The nodal acceleration \(\ddot{\varvec{d}}(t)\) can be approximated by the displacements, that is

$$\begin{aligned} \ddot{\varvec{d}}(t) = \frac{1}{\Delta t^2}\left[ \varvec{d}(t-\Delta t) - 2\varvec{d}(t) + \varvec{d}(t+\Delta t)\right] , \end{aligned}$$
(A.6)

where \(\Delta t\) denotes the time step. Substituting the approximation of the acceleration, (i.e., Eq. (A.6)) into Eq. (A.5) yields

$$\begin{aligned} \frac{\varvec{M}}{\Delta t^2}\varvec{d}(t+\Delta t) = \varvec{F}(t) - \left( \varvec{K}-\frac{2\varvec{M}}{\Delta t^2}\right) {\varvec{d}}(t) - \frac{\varvec{M}}{\Delta t^2}\varvec{d}(t-\Delta t). \end{aligned}$$
(A.7)

According to the above Yes, we have explained that it is an approximation, but in order to facilitate the derivation subsequently, we still use the expression of equality., the displacement solution at time \(t+\Delta t\) can be solved once the displacements at the last two steps, i.e., \({\varvec{d}}(t)\) and \(\varvec{d}(t-\Delta t)\), have been obtained.

Note that we need to consider the initial conditions for Eq. (A.7). That is, when \(t=0\), we need both an initial displacement \({\varvec{d}}(0)\) and a value of \({\varvec{d}}(-\Delta t)\) in order to calculate the next step displacement \({\varvec{d}}(\Delta t)\). According to the central difference approximation, a nodal velocity can be expressed as follows:

$$\begin{aligned} \dot{\varvec{d}}(t) = \frac{1}{2\Delta t}\left[ -\varvec{d}(t-\Delta t) + \varvec{d}(t+\Delta t)\right] . \end{aligned}$$
(A.8)

Considering both Eqs. (A.6) and (A.8), when \(t=0\), the following equation holds:

$$\begin{aligned} {\varvec{d}}(-\Delta t) = {\varvec{d}}(0) - \Delta t\dot{\varvec{d}}(0) + \frac{\Delta t^2}{2}\ddot{\varvec{d}}(0), \end{aligned}$$
(A.9)

where \(\ddot{\varvec{d}}(0)\) can be calculated via Eq. (A.5) when \(t=0\), i.e.,

$$\begin{aligned} \ddot{\varvec{d}}(0) = \varvec{M}^{-1}\left[ \varvec{F}(0)-\varvec{K}\varvec{d}(0)\right] . \end{aligned}$$
(A.10)

Consequently, the initial velocity \(\dot{\varvec{d}}(0)\) is necessary to replace the value \({\varvec{d}}(-\Delta t)\) when \(t=0\). Thus, both displacement \({\varvec{d}}(0)\) and velocity \(\dot{\varvec{d}}(0)\) are the initial conditions for Eq. (A.7).

1.3 Appendix A.3: Numerical algorithm procedure

We rewrite Eq. (A.7) into a form of linear equation as follows:

$$\begin{aligned} \hat{\varvec{M}}\varvec{d}_{t+\Delta t} = \varvec{F}_t, \end{aligned}$$
(A.11)

where \(\hat{\varvec{M}}=\varvec{M}/\Delta t^2\), \(\varvec{d}_{t+\Delta t}=\varvec{d}(t+\Delta t)\), and \(\varvec{F}_t=\varvec{F}(t) - \left( \varvec{K}-2\hat{\varvec{M}}\right) {\varvec{d}}(t) - \hat{\varvec{M}}\varvec{d}(t-\Delta t)\). Then, the algorithm procedure to solve the hybrid equation of motion is stated as follows:

  1. 1.

    Assemble mass matrix \(\varvec{M}\) and stiffness matrix \(\varvec{K}\).

  2. 2.

    Set the initial displacement \(\varvec{d}(0)\), velocity \(\dot{\varvec{d}}(0)\), and external force \(\varvec{F}(0)\).

  3. 3.

    Calculate the initial acceleration \(\ddot{\varvec{d}}(0)\) in terms of Eq. (A.10).

  4. 4.

    Select the time step \(\Delta t\), satisfying \(\Delta t<\Delta t_c\), where \(\Delta t_c\) denotes the critical value to ensure the stability of solution.

  5. 5.

    Calculate \({\varvec{d}}(-\Delta t)\) in terms of Eq. (A.9).

  6. 6.

    Calculate \(\hat{\varvec{M}}=\varvec{M}/\Delta t^2\).

  7. 7.

    For every time step (\(t=0, \Delta t, 2\Delta t, \cdots \)), calculate the effective loading \(\varvec{F}_t=\varvec{F}(t) - \left( \varvec{K}-2\hat{\varvec{M}}\right) {\varvec{d}}(t) - \hat{\varvec{M}}\varvec{d}(t-\Delta t)\).

  8. 8.

    Solve the linear equation Eq. (A.11), to get the displacement solution \(\varvec{d}_{t+\Delta t}\) at time \(t+\Delta t\).

  9. 9.

    If needed, calculate acceleration \(\ddot{\varvec{d}}(t)\) and velocity \(\dot{\varvec{d}}(t)\) at time t in terms of Eqs. (A.6) and (A.8).

  10. 10.

    If t is less than the final time, then \(t=t+\Delta t\) and return to the step 7. Otherwise, end the calculation.

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Han, F., Liu, S. & Lubineau, G. A dynamic hybrid local/nonlocal continuum model for wave propagation. Comput Mech 67, 385–407 (2021). https://doi.org/10.1007/s00466-020-01938-7

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