## 1 Introduction  
Topology optimization (TO) is a powerful computational design methodology that seeks optimal material distribution within a given domain, subject to physics-based constraints and objectives [1, 2]. Traditionally applied to single-material structures, TO has enabled the synthesis of highly efficient, lightweight, and innovative components across engineering disciplines. However, as manufacturing technologies such as additive manufacturing and multi-material printing have advanced [3], there is a growing need to extend TO methodologies to accommodate multiple material phases. This demand is driven by the potential of multimaterial structures to offer superior performance by strategically combining the desirable properties of different materials--e.g., stiffness, energy absorption, heat conduction, or cost-efficiency--in a single optimized layout. Multimaterial Topology Optimization (MMTO) introduces several new challenges that distinguish it from its single-material counterpart. First, the interpolation of material properties becomes substantially more complex. Unlike the conventional SIMP (Solid Isotropic Material with Penalization) scheme [2] which interpolates between solid and void, MMTO must handle transitions across multiple discrete materials without producing physically meaningless mixtures or gray zones. Techniques such as the Discrete Material Optimization (DMO) [4], peak functions [5], and ordered SIMP interpolations [3] have been developed to address these challenges. More recent efforts have focused on improving the manufacturability and computational efficiency of MMTO. [PERSON] et al. [6] proposed a mapping-based interpolation function implemented in a compact MATLAB code, extending the educational 88-line TO code [7] to handle 2D and 3D multimaterial structures. Their approach yields clear 0-1 solutions and ensures discrete material regions suitable for fabrication. Similarly, [PERSON] et al. [8] introduced a single-variable-based interpolation function with continuous derivatives to enhance optimization convergence and reduce design dimensionality. A significant issue in MMTO is phase mixing at material boundaries, which complicates both physical interpretation and manufacturing. [PERSON] et al. [9] tackled this by developing a multi-phase topology optimization framework with independent design variables for each phase. Their approach employs Heaviside projections with phase-specific length-scale control, aggregated constraints to avoid overfilling, and consistent penalization for phase separation. This method enables large-scale optimization and mitigates the tendency for undesired material interpenetration. Beyond mechanical compliance, MMTO has also been extended to address more complex physical behaviors such as plasticity and transient thermal conduction. [PERSON] et al. [10] introduced a multiobjective MMTO framework for elastoplastic composite structures that simultaneously enhances stiffness, delays yielding, and maximizes energy dissipation. Their use of return mapping algorithms and history-dependent adjoint sensitivities marks an important advance in integrating realistic material behavior into TO. On the thermal side, [PERSON] et al. [4] applied the DMO method to MMTO for unsteady heat conduction, emphasizing its robustness against material ordering and its capability to optimize thermal pathways in time-dependent problems. Several common threads emerge across these studies. First, interpolation strategies remain a central concern: whether using multi-variable SIMP [11], DMO [12], or peak/ordered functions [3; 5], each method seeks to balance fidelity, computational cost, and manufacturability.