## 2 Method
### Adaptive Stress-Strain Optimization Strategy (ASSOS)
#### 2.4.1 Iterative Error Minimization  
ASSOS adopts a robust feedback-driven iterative methodology to systematically minimize residual stress imbalances, ensuring convergence towards physically consistent equilibrium states. At the core of this process lies the computation of the residual error:  
\[\mathbf{R}=\mathbf{\
abla}\cdot\mathbf{\sigma}-\mathbf{f}, \tag{26}\]  
where \(\mathbf{R}\) quantifies deviations from the equilibrium condition \(\mathbf{\
abla}\cdot\mathbf{\sigma}=\mathbf{f}\). This residual serves as a driving force for iterative correction, and the stress field is updated at each iteration according to:  
\[\sigma_{n+1}=\mathbf{\sigma}_{n}-\alpha\mathbf{R}_{n}, \tag{27}\]  
where \(\alpha\) is an adaptive step size, dynamically computed via a line search strategy to maximize the reduction of \(\mathbf{R}\) while maintaining stability. For problems involving nonlinear material behavior, this update is extended using an incremental formulation:  
\[\Delta\mathbf{\sigma}=\mathbf{C}:\Delta\mathbf{\varepsilon}, \tag{28}\]  
where \(\Delta\mathbf{\sigma}\) and \(\Delta\mathbf{\varepsilon}\) are the stress and strain increments, respectively, and \(\mathbf{C}\) is the tangent stiffness tensor reflecting the material's instantaneous response. This approach ensures that path-dependent phenomena, such as plastic deformation or viscoelastic creep, are accurately captured.  
Figure 3: The Adaptive Stress–Strain Optimization Strategy (ASSOS) architecture integrates multi-scale coupling with shared MLP layers for feature extraction across scales \(S_{1}\) and \(S_{2}\). Key operations include MaxPooling (P), repetition (R), and concatenation (\(\oplus\)), enabling efficient stress–strain representation and fusion. ASSOS leverages shared representations to balance computational efficiency with accurate modeling of heterogeneous material behaviors.  
To enhance the stability and convergence of the iterative process, energy principles are incorporated. The total potential energy \(\mathcal{E}\) of the system is minimized at each step, defined as  
\[\mathcal{E}=\int_{\Omega}\left(\frac{1}{2}\sigma:\varepsilon-\mathbf{f}\cdot \mathbf{u}\right)d\Omega, \tag{29}\]  
where \(\varepsilon\) is the strain tensor, \(\mathbf{u}\) is the displacement vector, and \(\Omega\) denotes the domain of interest. The energy gradient drives corrections:  
\[\Delta\mathcal{E}=\frac{\partial\mathcal{E}}{\partial\sigma}:\Delta\sigma, \tag{30}\]  
ensuring that the iterative process does not introduce instabilities or violate thermodynamic consistency.  
Additionally, ASSOS employs adaptive relaxation techniques to manage high-gradient regions, where abrupt stress variations might hinder convergence. A relaxation factor \(\beta\) is introduced:  
\[\sigma_{n+1}=\sigma_{n}-\beta\alpha\mathbf{R}_{n}, \tag{31}\]  
where \(\beta\in(0,1]\) adjusts the correction magnitude, facilitating stable convergence in challenging scenarios such as near crack tips or phase boundaries.  
To monitor and ensure convergence, a dual residual criterion is employed. The first ensures that the norm of the residual stress falls below a tolerance:  
\[\|\mathbf{R}\|<\varepsilon, \tag{32}\]  
and the second enforces that the change in stress between iterations satisfies:  
\[\|\sigma_{n+1}-\sigma_{n}\|<\delta, \tag{33}\]  
where \(\varepsilon\) and \(\delta\) are user-defined thresholds.