_Algorithms_**2009**, \(2\), 1281-1300; doi:10.3390/a2041281  
_Article_  
**An Adaptive \(h\)-Refinement Algorithm for Local Damage Models**  
**[PERSON] and [PERSON]**\({}^{\star}\)  
Department of Engineering Science and Mechanics, The Pennsylvania State University, 212 EES Building, University Park, PA 16802, USA; E-mail: [EMAIL_ADDRESS]  
\({}^{\star}\) Author to whom correspondence should be addressed; E-mail: [EMAIL_ADDRESS]. _Received: 30 April 2009; in revised form: 18 September 2009 / Accepted: 28 September 2009 / Published: 6 October 2009_  
**Abstract:** An adaptive mesh refinement strategy is proposed for local damage models that often arise from internal state variable based continuum damage models. The proposed algorithm employs both the finite element method and the finite difference method to integrate the equations of motion of a linear elastic material with simple isotropic microcracking. The challenges of this problem include the time integration of coupled partial differential equations with time-dependent coefficients, and the proper choice of solution spaces to yield a stable finite element formulation. Discontinuous elements are used for the representation of the damage field, as it is believed that this reduction in regularity is more consistent with the physical nature of evolving microcracking. The adaptive mesh refinement algorithm relies on custom refinement indicators, two of which are presented and compared. The two refinement indicators we explore are based on the time rate of change of the damage field and on the energy release rate, respectively, where the energy release rate measures the energy per unit volume available for damage to evolve. We observe the performance of the proposed algorithm and refinement indicators by comparing the predicted damage morphology on different meshes, hence judging the capability of the proposed technique to address, but not eliminate, the mesh dependency present in the solutions of the damage field. **Keywords:** adaptive mesh refinement; damage-based refinement indicators; dynamic fracture; discontinuous-Galerkin finite element methods; continuum damage mechanics Introduction  
Continuum damage mechanics (CDM) failure models are well-known and implemented extensively in many engineering applications. The attraction to these models is partly based on the use of macroscopic fields to quantify the state of the microstructure of a material, thus allowing for damage models to be easily incorporated into existing commercial finite element analysis software. A large body of literature exists on the development and physical basis of continuum damage models; however, in this paper, we do not seek to add to it. Rather, we explore the use of adaptive mesh refinement techniques to address a common issue in the implementation of continuum damage models. Damage models derived in the framework of continuum mechanics with internal state variables often employ evolution equations for the damage field(s) that are ordinary differential equations (ODEs) in time, and are thus without the action of any spatial differential operators on the damage field. The lack of these spatial differential operators induces localization and severe mesh dependence (see, e.g., [1]) in the damage field. When not relying on a reformulation of the evolution equation(s), a simple approach to dealing with the localization problem is to define a physically-based minimum element size. However, even when this strategy is chosen, mesh dependence of the solution persists, and the level of mesh refinement necessary for resolving the damage field may be beyond the computational resources available. With this in mind, the main goal of this paper is to begin to explore whether or not, after having chosen a physically-based minimum element size, adaptive mesh refinement may offer an effective strategy to reduce the problem size while still resolving the damage morphology.