Dual-horizon peridynamics

Peridynamics is a theoretical framework within the field of continuum mechanics that is specifically designed to address deformations characterized by discontinuities, with a particular emphasis on fractures. Uniformity of the horizon is a typical prerequisite for all material points. The concept of dual-horizon is proposed in dual-horizon peridynamics to overcome the limitation of a fixed horizon, and the governing equations are reformulated accordingly.

Horizon and dual-horizon

The fundamental principles underlying dual-horizon peridynamics pertain to the notions of horizon and dual-horizon.

File:Dual-horizon peridynamics.svg

Figure 1. Fixed horizon peridynamics to dual-horizon peridynamics with variable horizons

Horizon of $\mathbf{𝐱}$ is a finite-size neighborhood of material point $\mathbf{𝐱}$. $$H_{\mathbf{𝐱}}=\{{\mathbf{𝐱}}^{\prime }|||{\mathbf{𝐱}}^{\prime }-\mathbf{𝐱}||\le \delta \}$$ where $\delta$ is the radius of the neighborhood.

Dual-horizon is defined as a union of points whose horizons include $\mathbf{𝐱}$, denoted by $$H{\text{'}}_{\mathbf{𝐱}}=\{{\mathbf{𝐱}}^{\prime }|\mathbf{𝐱}\in H_{{\mathbf{𝐱}}^{\prime }}\}$$ The definition of dual-horizon implies ${\mathbf{𝐱}}^{\prime }\in H{\text{'}}_{\mathbf{𝐱}}\leftrightarrow \mathbf{𝐱}\in H_{{\mathbf{𝐱}}^{\prime }}$.

The governing equations of conventional peridynamics $$\underset{H_{\mathbf{𝐱}}}{\int }({\mathbf{𝐟}}_{\mathbf{𝐱}{\mathbf{𝐱}}^{\prime }}-{\mathbf{𝐟}}_{{\mathbf{𝐱}}^{\prime }\mathbf{𝐱}})d{V}_{{\mathbf{𝐱}}^{\prime }}+\mathbf{𝐛}=\rho {\ddot{\mathbf{𝐮}}}_{\mathbf{𝐱}}$$ where ${\mathbf{𝐟}}_{\mathbf{𝐱}{\mathbf{𝐱}}^{\prime }}$ is the bond-force density between material point $\mathbf{𝐱}$ and ${\mathbf{𝐱}}^{\prime }$. The body force density is represented by the symbol $\mathbf{𝐛}$, while the density is denoted by $\rho$. The displacement field is denoted by the symbol $\mathbf{𝐮}$.

The governing equations of dual-horizon peridynamics are $$\underset{H_{\mathbf{𝐱}}}{\int }{\mathbf{𝐟}}_{\mathbf{𝐱}{\mathbf{𝐱}}^{\prime }}d{V}_{{\mathbf{𝐱}}^{\prime }}-\underset{H_{{\mathbf{𝐱}}^{\prime }}}{\int }{\mathbf{𝐟}}_{{\mathbf{𝐱}}^{\prime }\mathbf{𝐱}}d{V}_{{\mathbf{𝐱}}^{\prime }}+\mathbf{𝐛}=\rho {\ddot{\mathbf{𝐮}}}_{\mathbf{𝐱}}$$ The dual-horizon peridynamics is a straightforward adaptation of the traditional peridynamics. It facilitates the use of peridynamics for non-uniform horizons in all material points. The present configuration bears resemblance to the non-uniform discretization technique employed in finite element methods, thereby leading to a substantial enhancement in computational efficacy.

Peridynamics versus dual-horizon peridynamics

The dual-horizon peridynamics encompasses the conventional constant-horizon peridynamics as a specific instance. The table below outlines a comparison between peridynamics and dual-horizon peridynamics.

Peridynamics vs. dual-horizon peridynamics

Model	Peridynamics	Dual-horizon peridynamics
Governing equations	$\underset{H_{\mathbf{𝐱}}}{\int }({\mathbf{𝐟}}_{\mathbf{𝐱}{\mathbf{𝐱}}^{\prime }}-{\mathbf{𝐟}}_{{\mathbf{𝐱}}^{\prime }\mathbf{𝐱}})d{V}_{{\mathbf{𝐱}}^{\prime }}+\mathbf{𝐛}=\rho {\ddot{\mathbf{𝐮}}}_{\mathbf{𝐱}}$	$\underset{H_{\mathbf{𝐱}}}{\int }{\mathbf{𝐟}}_{\mathbf{𝐱}{\mathbf{𝐱}}^{\prime }}d{V}_{{\mathbf{𝐱}}^{\prime }}-\underset{H_{{\mathbf{𝐱}}^{\prime }}}{\int }{\mathbf{𝐟}}_{{\mathbf{𝐱}}^{\prime }\mathbf{𝐱}}d{V}_{{\mathbf{𝐱}}^{\prime }}+\mathbf{𝐛}=\rho {\ddot{\mathbf{𝐮}}}_{\mathbf{𝐱}}$
BB-PD	$\begin{array}{l}{\mathbf{𝐟}}_{{\mathbf{𝐱}\mathbf{𝐱}}^{\prime }}-{\mathbf{𝐟}}_{{\mathbf{𝐱}}^{\prime }\mathbf{𝐱}}=c{s}_{{\mathbf{𝐱}\mathbf{𝐱}}^{\prime }}\mathbf{𝐧},\mathrm{\forall }{\mathbf{𝐱}}^{\prime }\in H_{\mathbf{𝐱}}\end{array}$	$\begin{array}{l}{\mathbf{𝐟}}_{\mathbf{𝐱}{\mathbf{𝐱}}^{\prime }}=C\left({\delta }_{\mathbf{𝐱}}\right)s_{{\mathbf{𝐱}\mathbf{𝐱}}^{\prime }}\mathbf{𝐧},\mathrm{\forall }{\mathbf{𝐱}}^{\prime }\in H_{\mathbf{𝐱}}\\ {\mathbf{𝐟}}_{{\mathbf{𝐱}}^{\prime }\mathbf{𝐱}}=C\left({\delta }_{{\mathbf{𝐱}}^{\prime }}\right)s_{{\mathbf{𝐱}\mathbf{𝐱}}^{\prime }}(-\mathbf{𝐧}),\mathrm{\forall }{\mathbf{𝐱}}^{\prime }\in H_{{\mathbf{𝐱}}^{\prime }}\end{array}$
OSB-PD	$\begin{array}{l}{\mathbf{𝐟}}_{{\mathbf{𝐱}\mathbf{𝐱}}^{\prime }}=\underset{\_}{t}⟨\mathit{\xi }⟩\mathbf{𝐧}\\ {\mathbf{𝐟}}_{{\mathbf{𝐱}}^{\prime }\mathbf{𝐱}}=\underset{\_}{t^{\prime }}⟨-\mathit{\xi }⟩(-\mathbf{𝐧}),\mathrm{\forall }{\mathbf{𝐱}}^{\prime }\in H_{\mathbf{𝐱}}\end{array}$	$\begin{array}{l}{\mathbf{𝐟}}_{{\mathbf{𝐱}\mathbf{𝐱}}^{\prime }}=\underset{\_}{t}⟨\mathit{\xi }⟩\mathbf{𝐧},\mathrm{\forall }{\mathbf{𝐱}}^{\prime }\in H_{\mathbf{𝐱}}\\ {\mathbf{𝐟}}_{{\mathbf{𝐱}}^{\prime }\mathbf{𝐱}}={\underset{\_}{t}}^{\prime }⟨-\mathit{\xi }⟩(-\mathbf{𝐧}),\mathrm{\forall }{\mathbf{𝐱}}^{\prime }\in H_{{\mathbf{𝐱}}^{\prime }}\end{array}$
NOSB-PD	$\begin{array}{l}{\mathbf{𝐟}}_{\mathbf{𝐱}{\mathbf{𝐱}}^{\prime }}=\underset{\_}{\mathbf{𝐓}}[{\mathbf{𝐱}}^{\prime }-\mathbf{𝐱}],\\ {\mathbf{𝐟}}_{{\mathbf{𝐱}}^{\prime }\mathbf{𝐱}}={\underset{\_}{\mathbf{𝐓}}}^{\prime }[\mathbf{𝐱}-{\mathbf{𝐱}}^{\prime }],\mathrm{\forall }{\mathbf{𝐱}}^{\prime }\in H_{\mathbf{𝐱}}\end{array}$	$\begin{array}{l}{\mathbf{𝐟}}_{\mathbf{𝐱}{\mathbf{𝐱}}^{\prime }}=\underset{\_}{\mathbf{𝐓}}[{\mathbf{𝐱}}^{\prime }-\mathbf{𝐱}],\mathrm{\forall }{\mathbf{𝐱}}^{\prime }\in H_{\mathbf{𝐱}}\\ {\mathbf{𝐟}}_{{\mathbf{𝐱}}^{\prime }\mathbf{𝐱}}={\underset{\_}{\mathbf{𝐓}}}^{\prime }[\mathbf{𝐱}-{\mathbf{𝐱}}^{\prime }],\mathrm{\forall }{\mathbf{𝐱}}^{\prime }\in H_{{\mathbf{𝐱}}^{\prime }}\end{array}$

where $\mathbf{𝐧}=\frac{\mathit{\eta }+\mathit{\xi }}{\Vert \mathit{\eta }+\mathit{\xi }\Vert },\mathit{\eta }={\mathbf{𝐮}}^{\prime }-\mathbf{𝐮},\mathit{\xi }={\mathbf{𝐱}}^{\prime }-\mathbf{𝐱}$.

The three peridynamics models under consideration are bond-based peridynamics (BB-PD), state-based peridynamics (OSB-PD), and nonordinary state-based peridynamics (NOSB-PD), each with their respective abbreviations.

Comment on numerical implementation

Despite the introduction of the dual-horizon, the numerical implementation did not incur any additional expenses. The dual-horizon bond forces can be automatically satisfied during the computation of forces from the horizons of other material points. This concept is demonstrated through a straightforward illustration.

Force accumulation in dual-horizon peridynamics

For example: $H_{{\mathbf{𝐱}}_0}=\{{\mathbf{𝐱}}_1,{\mathbf{𝐱}}_2,{\mathbf{𝐱}}_4,{\mathbf{𝐱}}_6\},H{\text{'}}_{{\mathbf{𝐱}}_0}=\{{\mathbf{𝐱}}_1,{\mathbf{𝐱}}_2,{\mathbf{𝐱}}_3,{\mathbf{𝐱}}_4\}$ in Figure 1.

The bond force density on a bond between two material points ${\mathbf{𝐱}}_i$ and ${\mathbf{𝐱}}_j$ is denoted by ${\mathbf{𝐟}}_{ij}$. The symbol $\mathrm{\Delta }{V}_i$ represents the volume of a specific material point denoted by ${\mathbf{𝐱}}_i$.

Internal forces from $H_{{\mathbf{𝐱}}_0}$

• Add ${\mathbf{𝐟}}_{01}\mathrm{\Delta }{V}_0\mathrm{\Delta }{V}_1$ to ${\mathbf{𝐱}}_0$, add reaction force $-{\mathbf{𝐟}}_{01}\mathrm{\Delta }{V}_0\mathrm{\Delta }{V}_1$ to ${\mathbf{𝐱}}_1$
• Add ${\mathbf{𝐟}}_{02}\mathrm{\Delta }{V}_0\mathrm{\Delta }{V}_2$ to ${\mathbf{𝐱}}_0$, add reaction force $-{\mathbf{𝐟}}_{02}\mathrm{\Delta }{V}_0\mathrm{\Delta }{V}_2$ to ${\mathbf{𝐱}}_2$
• Add ${\mathbf{𝐟}}_{04}\mathrm{\Delta }{V}_0\mathrm{\Delta }{V}_4$ to ${\mathbf{𝐱}}_0$, add reaction force $-{\mathbf{𝐟}}_{04}\mathrm{\Delta }{V}_0\mathrm{\Delta }{V}_4$ to ${\mathbf{𝐱}}_4$
• Add ${\mathbf{𝐟}}_{06}\mathrm{\Delta }{V}_0\mathrm{\Delta }{V}_6$ to ${\mathbf{𝐱}}_0$, add reaction force $-{\mathbf{𝐟}}_{06}\mathrm{\Delta }{V}_0\mathrm{\Delta }{V}_6$ to ${\mathbf{𝐱}}_6$

Internal forces from $H{\text{'}}_{{\mathbf{𝐱}}_0}$ are calculated when calculating bond forces in $H_{{\mathbf{𝐱}}_1},H_{{\mathbf{𝐱}}_2},H_{{\mathbf{𝐱}}_3},H_{{\mathbf{𝐱}}_4}$.

• In $H_{{\mathbf{𝐱}}_1}$, add ${\mathbf{𝐟}}_{10}\mathrm{\Delta }{V}_1\mathrm{\Delta }{V}_0$ to ${\mathbf{𝐱}}_1$, add reaction force $-{\mathbf{𝐟}}_{10}\mathrm{\Delta }{V}_1\mathrm{\Delta }{V}_0$ to ${\mathbf{𝐱}}_0$
• In $H_{{\mathbf{𝐱}}_2}$, add ${\mathbf{𝐟}}_{20}\mathrm{\Delta }{V}_2\mathrm{\Delta }{V}_0$ to ${\mathbf{𝐱}}_2$, add reaction force $-{\mathbf{𝐟}}_{20}\mathrm{\Delta }{V}_2\mathrm{\Delta }{V}_0$ to ${\mathbf{𝐱}}_0$
• In $H_{{\mathbf{𝐱}}_3}$, add ${\mathbf{𝐟}}_{30}\mathrm{\Delta }{V}_3\mathrm{\Delta }{V}_0$ to ${\mathbf{𝐱}}_3$, add reaction force $-{\mathbf{𝐟}}_{30}\mathrm{\Delta }{V}_3\mathrm{\Delta }{V}_0$ to ${\mathbf{𝐱}}_0$
• In $H_{{\mathbf{𝐱}}_4}$, add ${\mathbf{𝐟}}_{40}\mathrm{\Delta }{V}_4\mathrm{\Delta }{V}_0$ to ${\mathbf{𝐱}}_4$, add reaction force $-{\mathbf{𝐟}}_{40}\mathrm{\Delta }{V}_4\mathrm{\Delta }{V}_0$ to ${\mathbf{𝐱}}_0$

See also

• Fracture mechanics
• Continuum mechanics
• Peridynamics

External links

• Peridigm, an open-source computational peridynamics code
• website on peridynamics

This article "Dual-horizon peridynamics" is from Wikipedia. The list of its authors can be seen in its historical and/or the page Edithistory:Dual-horizon peridynamics. Articles copied from Draft Namespace on Wikipedia could be seen on the Draft Namespace of Wikipedia and not main one.

📰 Article(s) of the same category(ies)

• KOMPAS-3D
• Oseen–Frank limit
• Ensemble (fluid mechanics)
• Repeated superposition of finite strains
• Visual-CFD
• Tribo-fatigue
• Virtual element method

---