# An Adaptive h-Refinement Algorithm for Local Damage Models

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## Abstract

**:**

## 1. Introduction

## 2. Linear Elasticity with Microcracking

- $\dot{\tilde{\varphi}}=0$: Clearly, $\frac{\partial \tilde{\psi}}{\partial \varphi}\dot{\tilde{\varphi}}\le 0$ is trivially satisfied.
- $\dot{\tilde{\varphi}}>0$: By Equation (7), $G>{G}_{cr}>0$. By the definition of the energy release rate in (6), $\frac{\partial \tilde{\psi}}{\partial \varphi}<0$, and hence $\frac{\partial \tilde{\psi}}{\partial \varphi}\dot{\tilde{\varphi}}\le 0$ is satisfied.

## 3. Integration of Equations

#### 3.1. Finite Element Method

#### 3.2. Finite Difference Method

- Solve explicitly for ${\Phi}^{j,n+1}={\mathcal{F}}_{1}({\Phi}^{j,n};{U}^{j,n},{V}^{j,n})$ using the first order accurate forward Euler method, where ${\mathcal{F}}_{1}$ is $\mathcal{F}$ when $\dot{u}$ and $\dot{v}$ are set to zero.Specifically, the system to be discretized is:$${M}^{0}{\dot{\Phi}}^{j}={F}_{\varphi}.$$$$\begin{array}{cc}\hfill {M}_{ij}^{0}{\Phi}^{j,n+1}& ={M}_{ij}^{0}{\Phi}^{j,n}+\Delta t{F}_{\varphi}^{n}\hfill \end{array}$$$$\begin{array}{cc}& ={M}_{ij}^{0}{\Phi}^{j,n}+\Delta t\phantom{\rule{4pt}{0ex}}\left({\eta}_{c}^{n}\u2329\frac{\beta}{2}{(1-{\varphi}_{h}^{n})}^{\beta -1}\mathsf{C}\left[\nabla {\mathbf{u}}_{h}^{n}\right]\xb7\nabla {\mathbf{u}}_{h}^{n}-{G}_{cr}^{n}\u232a,{w}^{i}\right).\hfill \end{array}$$
- Solve implicitly for $[{U}^{j,n+1},{V}^{j,n+1}]={\mathcal{F}}_{2}\left({U}^{j,n+1},{V}^{j,n+1};{U}^{j,n},{V}^{j,n},{\Phi}^{j,n+1},{\Phi}^{j,n}\right)$, where ${\mathcal{F}}_{2}$ is $\mathcal{F}$ when $\dot{\varphi}$ is set to zero.

`deal.ii`finite element library (see, for reference, [14]).

## 4. Adaptive Mesh Refinement Algorithm

## 5. Numerical Experiments

**Figure 3.**Geometry and boundary conditions of a two-dimensional bar fixed at ${X}_{1}=0$ and subject to an applied traction at ${X}_{1}=L$.

Elastic Modulus | 9 ×10^{9} Pa | Density | 1.7 × 10^{3} kg/m^{3} |

Poisson Ratio | 0.3 | G_{cr} | 3 × 10^{7} J/m^{3} |

η_{c} | 0.1 m^{3}/J·s | β | 2 |

τ | 20 μs | s_{x} | 5× 10^{7} Pa |

**Figure 4.**Damage-based refinement scheme: from top to bottom, triangulations of the two-dimensional bar at times $t=0$, $t=$88.5 μs, $t=$ 106.1 μs, and $t=$ 115.9 μs. The refinement is concentrated at the fixed end of the bar, which is where we expect the damage to increase.

**Figure 5.**Damage based refinement: from left to right, zoom-in images of the damage at the fixed-end of the bar $X=0$, at $t=$88.5 μs, $t=$ 106.1 μs, and $t=$ 115.9 μs. The damage nucleates at the corners of the bar, and progresses towards the center.

**Figure 6.**Energy release rate based refinement, with $c\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}1$: from left to right, zoom-in images of the damage at the fixed-end of the bar $X=0$, at $t=$88.5 μs, $t=$ 106.1 μs, and $t=$ 115.9 μs.

**Figure 7.**Energy release rate based refinement, $c\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}0.5$: The same simulation as in Figure 6, but now with the refinement indicator parameter $c=0.5$. Note the difference in the mesh pattern, specifically the refinement of cells in which damage is not evolving.

**Figure 8.**The first principle stress versus the first principle strain at a point which becomes fully damaged (left) and the corresponding time series of the damage variable and Helmholtz free energy density (right). Dynamic effects can be observed in both data sets. On the right, region I corresponds to the response before damage evolution, region II to the brief period of damage evolution, and region III to the period when the material is fully damaged.

#### 5.1. Idealized Compact Test Specimen

**Figure 9.**Geometry and initial meshes for the simulations of an idealized 2D compact test specimen. Vertical traction loading is applied to the faces denoted as ${\Gamma}_{N}$, creating a situation likely to result in Mode I failure. The meshes were created using the pave (left) and submap (right) methods in Cubit [17].

**Figure 10.**Damage solutions on two very different meshes with the two proposed refinement algorithms. In these simulations, the constant in (39) is $c=0.5$, leading to the preemptive refinement of non-damaged cells in Figure 10(c) and Figure 10(d). In general, the solutions are all similar, thus we are encouraged that adaptive mesh refinement may be able to help control mesh-dependency issues; however, it is clear that the extent of localization is different between the two meshes and not yet to a point of mathematical comparison.

**Figure 11.**Zoom-in of the damage solutions in Figure 10(a) and Figure 10(b), respectively, showing the damage field solution produced by the damage-based refinement algorithm. The extent to which the damage is localized is clearly visible, and, for this first attempt, is considered thus far to be sucessful.

## 6. Summary

## Acknowledgements

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^{1.}The notation ${H}^{1}$ is standard in the field of partial differential equations to denote the Sobolev space of Lebesgue integrable functions whose weak derivatives of order one are also integrable. The notation ${L}^{2}$ denotes the space of square (Lebesgue) integrable functions. Although we have not explicitly indicated it, it is understood that the ${H}^{1}$ spaces in question conform to the prescribed Dirichlet boundary conditions for the fields of interest.^{2.}Ω is the domain of the problem.^{3.}This is typically the case in Discontinous Galerkin (DG) finite element methods (see, e.g., [11]).^{4.}Note that there is a distinction between error estimators and refinement indicators. The former measure the error present in the approximate solution for a given cell in the triangulation, while the latter simply determines whether or not the cell should be refined. While these two concepts are often used together, in this paper, we are simply interested in refining the mesh where damage is evolving without any direct attempt to estimate the error.

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**MDPI and ACS Style**

Pitt, J.S.; Costanzo, F.
An Adaptive *h*-Refinement Algorithm for Local Damage Models. *Algorithms* **2009**, *2*, 1281-1300.
https://doi.org/10.3390/a2041281

**AMA Style**

Pitt JS, Costanzo F.
An Adaptive *h*-Refinement Algorithm for Local Damage Models. *Algorithms*. 2009; 2(4):1281-1300.
https://doi.org/10.3390/a2041281

**Chicago/Turabian Style**

Pitt, Jonathan S., and Francesco Costanzo.
2009. "An Adaptive *h*-Refinement Algorithm for Local Damage Models" *Algorithms* 2, no. 4: 1281-1300.
https://doi.org/10.3390/a2041281