## 2 Method
### Adaptive Stress-Strain Optimization Strategy (ASSOS)
#### 2.4.2 Adaptive Mesh Refinement and Multi-Scale Coupling  
ASSOS employs an advanced adaptive mesh refinement (AMR) strategy to efficiently resolve localized stress concentrations and other high-gradient regions, such as crack tips, inclusions, or phase boundaries (as shown in Figure 4). The refinement criterion is governed by element-wise error estimations:  
\[\eta_{\varepsilon}=\int_{\Omega_{\varepsilon}}\|\mathbf{\
abla}\cdot\mathbf{\sigma}- \mathbf{f}\|^{2}\,d\Omega_{\varepsilon}, \tag{34}\]  
where \(\eta_{\varepsilon}\) quantifies the residual error over each element \(\Omega_{\varepsilon}\). Elements with high \(\eta_{\varepsilon}\) values are flagged for refinement, dynamically increasing mesh resolution in critical regions. This ensures that computational resources are focused where they are most needed, thereby maintaining accuracy while minimizing computational costs. To further optimize computational efficiency, ASSOS employs hierarchical refinement, where coarse meshes are used in regions with low stress gradients, while finer meshes are applied selectively in areas exhibiting high errors. This process is iteratively updated based on a convergence criterion:  
\[\|\eta_{\varepsilon}\|_{\text{max}}<\epsilon_{\text{AMR}}, \tag{35}\]  
where \(\epsilon_{\text{AMR}}\) is a predefined tolerance for the residual error. In addition to AMR, ASSOS integrates multi-scale coupling techniques to accurately capture the heterogeneity and complexity of material behavior. Micro-scale simulations are performed to characterize local material properties, such as grain-level anisotropy, voids,or micro-cracks. These properties are then upscaled to inform the macro-scale model via homogenization:  
\[\mathbf{C}_{\mathrm{macro}}=\langle\mathbf{C}_{\mathrm{micro}}\rangle, \tag{36}\]  
where \(\mathbf{C}_{\mathrm{micro}}\) represents the local stiffness tensor at the micro-scale, and \(\langle\cdot\rangle\) is an averaging operator that can account for volume fractions, energy equivalence, or other homogenization schemes. To bridge the micro- and macro-scales effectively, ASSOS utilizes a two-way coupling mechanism. Information flows from the micro-scale to update macro-scale properties and from the macro-scale to define boundary conditions or loading scenarios for the micro-scale. This interaction is facilitated through nested iterations:  
\[\Delta\mathbf{u}_{\mathrm{macro}}\rightarrow\Delta\mathbf{u}_{\mathrm{micro}} \quad\text{and}\quad\Delta\mathbf{e}_{\mathrm{micro}}\rightarrow\Delta \mathbf{e}_{\mathrm{macro}}, \tag{37}\]  
where \(\Delta\mathbf{u}\) and \(\Delta\mathbf{e}\) represent incremental displacements and stresses, respectively. For highly heterogeneous materials, localized representative volume elements (RVEs) are used to extract microstructural responses. These RVEs are dynamically updated during simulation to account for evolving material states, such as damage accumulation or phase transformations. The effective macro-scale response is then recalibrated as  
\[\mathbf{C}_{\mathrm{macro}}(t)=\langle\mathbf{C}_{\mathrm{micro}}(t)\rangle, \tag{38}\]  
where \(t\) denotes the time or deformation state, ensuring that the macro-scale model remains consistent with micro-scale phenomena. Additionally, AMR is extended to handle evolving geometries, such as crack propagation or material erosion. Elements near advancing cracks are adaptively refined to capture the stress intensity factors (SIFs) accurately. The SIFs are computed using:  
\[K_{I}=\lim_{r\to 0}\sqrt{2\pi r}\boldsymbol{e}\cdot\mathbf{n}, \tag{39}\]  
where \(K_{I}\) represents the mode I SIF, \(r\) is the radial distance from the crack tip, and \(\mathbf{n}\) is the unit normal vector to the crack surface. This ensures that fracture mechanics are seamlessly integrated into the refinement process. Figure 4: Adaptive Mesh Refinement and Multi-Scale Coupling in ASSOS. This framework integrates adaptive mesh refinement (AMR) and multi-scale coupling to optimize computational efficiency while maintaining high accuracy in stress-strain predictions.