Design in Tiles (DiT)

Updated 22 December 2025

Design in Tiles (DiT) is a design and optimization paradigm that synthesizes complex systems through the parametric tiling of macro-domains using adaptable micro-tiles.
It reduces global complexity by systematically parameterizing, enumerating, and selecting tile primitives while enforcing interface compatibility and performance constraints.
DiT finds diverse applications in porous microstructures, phased arrays, metamaterials, and digital accelerators, demonstrating its versatility in addressing complex design challenges.

Design in Tiles (DiT) is a design and optimization paradigm in which complex structures, devices, or computational mappings are synthesized via the parametric tiling of a macro-domain by a set of micro-tiles whose geometry, topology, and material properties are themselves treated as variable, optimizable parameters. Originating in microarchitecture optimization for porous and graded materials, the DiT approach has since been extended to phased array synthesis, digital accelerator design, metamaterials, graphical texturing, and origami engineering. The unifying concept is the reduction of a global complexity problem to the systematic parameterization, enumeration, or combinatorial selection of local tile primitives, subject to interface compatibility and prescribed global objectives (Antolin et al., 2019).

1. Foundational Concepts and Tile Representation

Central to DiT is the notion of parameterized micro-tiles: small spatial (or abstract) domains governed by a vector of design variables, assembled in a regular or adaptive macro-grid to realize the target object or solution. A tile $\mathcal{M}$ is generally encoded as a trivariate B-spline or NURBS solid whose control points may be physically or functionally interpreted as follows (Antolin et al., 2019):

Geometric variables: wall thickness, pore size, channel curvature, overall aspect ratio
Topological variables: number of branches/bridges, connectivity (e.g. open cell vs. closed cell)
Material variables: spatial fields of conductivity $k(u,v,w)$ , Young’s modulus $E(u,v,w)$ , density $\rho(u,v,w)$

Control points may be extended into 4D or 5D to encode scalar/tensor fields relevant to local behavior. Crucially, micro-tiles are not fixed; the parameter vector $P = (p_1, \ldots, p_n)$ for a given tile varies over user-prescribed bounds $p_i^{\min} \leq p_i \leq p_i^{\max}$ chosen to guarantee a non-degenerate mapping.

The tile library comprises finitely many template topologies (e.g., tubes, truss elements, branching units), each associated with a parametric trivariate and endowed with matching boundaries (C⁰-continuity ensured by knot vector and control point identification). Discrete tile choices correspond to branching/fusion points in the global space of assembly (Antolin et al., 2019).

2. Global Optimization Formulation

Given an array $N_x \times N_y \times N_z$ of tiles covering the parametric domain $D$ (deformed into the physical macro-shape by a macro-map $T: D\to\mathbb{R}^3$ ), the full decision space is the concatenation $x = [P_{111}; \ldots; P_{N_xN_yN_z}] \in \mathbb{R}^{n\,N_\text{tiles}}$ . The global design objective is formulated as an unconstrained or constrained optimization problem:

Objective functions: sum performance metrics over tiles, for example,

$\text{maximize}_x\, Q(x) = \sum_{i,j,k} Q_\text{tile}(\mathcal{M}_{ijk}(P_{ijk}))$

(e.g., maximize heat exchange, minimize total weight, enforce prescribed response profile).

Constraints: stress limits, manufacturability ( $p_i\geq\text{min thickness}$ ), assembly continuity,

$g_j(x) \le 0,\quad h_k(x) = 0 \quad \forall j,k$

with $g_j$ for inequalities (e.g. $\sigma_\text{max}(x) - \sigma_\text{allowable} \le 0$ ), $h_k$ for equalities (continuity at interfaces).

The macro-framework admits arbitrary physics models—finite-element, lumped-parameter, or even analytic formulas—plugged into a black-box or gradient-free optimizer. For some applications, direct enumeration or analytic search is performed instead (Antolin et al., 2019).

A standard DiT optimization loop (cf. Algorithm 2, (Antolin et al., 2019)):

Initialize all tile parameters $P$
Assemble structure $\mathcal{M}(P) = \cup_{i,j,k} T(\mathcal{M}_{ijk}(P_{ijk}))$
Compute all metrics (performance, constraints)
Update $P\leftarrow\mathcal{O}(P;\text{metrics})$ where $\mathcal{O}$ is the optimization operator
Repeat until convergence

3. Interface Compatibility and Continuity

Tile-level compatibility is essential for robust assembly and manufacturability. The DiT paradigm achieves global C⁰-continuity via the following mechanisms (Antolin et al., 2019):

Matching knot structure and control points across shared tile faces, guaranteeing seamless physical and functional interface.
Hybrid approaches: mortaring (weak enforcement), overlapping/stitched trimmed surfaces for nonconforming face pairs.
Adaptive or variable tiling: when required, the framework supports adaptation by refining the tile grid where the objective gradients are high.

The entire porous object is assembled by enumerating all tile positions in $D$ , mapping each $M_{ijk}(P_{ijk})$ back through $T$ , and stitching the resulting trimmed surfaces along coincident knot lines (Antolin et al., 2019).

4. Numerical Optimization Strategies

Optimization over the tile parameter field $P(x,y,z)$ is application-driven:

Gradient-free (black-box) optimization: used for problems where the forward physics model is not differentiable or coupled; e.g., heat exchanger via lumped models, rocket grain design via analytical matching, or parameter sweeps in wing design.
Gradient-based/adjoint-coupled optimization: recognized as theoretically possible by propagating sensitivities through the parameterized geometry and physics solver, but not practically demonstrated for entire assemblies in current literature.
Specialized scheduling: in digital computing contexts (e.g., GEMM on manycore accelerators), DiT automates mapping and scheduling over tile grids using parameterizable executable models and search in high-dimensional schedule spaces, pruning infeasible configurations by performance models and cost criteria (Shen et al., 15 Dec 2025).

A generic pseudocode sketch (Algorithm 2, (Antolin et al., 2019)):

while not converged:
    for each tile (i,j,k):
        evaluate P_{ijk}, build M_{ijk}(P_{ijk})
    assemble structure = union of all T(M_{ijk})
    compute global metrics (performance, constraints)
    P = optimize_step(P, metrics)

5. Application Domains and Case Studies

The DiT methodology is broadly applicable and has been demonstrated in diverse domains:

Porous microstructures: DiT enables joint design of heat exchange efficiency and weight, with fine-tuned control at the tile level (e.g., channel spacing, wall thickness) and global constraints (e.g., stress) (Antolin et al., 2019).
Propulsion grain optimization: spatial distribution of accelerant/retardant fractions in tiles allows matching prescribed thrust-time profiles under simplified burn-front assumptions (Antolin et al., 2019).
Lightweight structures: wing design trades weight against stiffness by parameterizing wall thicknesses root-to-tip and skin-to-core, analyzed by isogeometric finite elements (~7M DOFs) (Antolin et al., 2019).
Metamaterial and robotic assembly: vertex-based Wang tile libraries are combined with combinatorial assembly and direct robot-assisted mold construction for rapid prototyping of mechanical metamaterials; performance metrics are validated experimentally (Doškář et al., 2023).
Graphics and texturing: dappled tiling and Wang tile-based microstructure synthesis are integrated into DiT to suppress periodic artifacts, support arbitrary large-scale nonrepetitive microstructures via level-set morphing algorithms and compatibility-encoded tile sets (Doškář et al., 2019, Kaji et al., 2016).
Phased arrays and digital accelerators: tile-based partitioning and scheduling are optimized to meet electromagnetic or computational performance specifications, with exact combinatorial enumeration (small arrays) or integer-coded genetic algorithms (large problems) operating over compressed representations of admissible tilings (Rocca et al., 2021, Rocca et al., 2021, Shen et al., 15 Dec 2025).

6. Limitations and Theoretical Extensions

DiT relies on several technical assumptions and faces practical constraints:

Solver coupling: Current methods primarily use black-box or uncoupled optimization; fully adjoint-based sensitivities are only proposed, not implemented, in global DiT pipelines.
Continuity: Enforcement is typically C⁰; higher-order smoothness (C¹, etc.) imposes stricter requirments on knot multiplicities and is rarely realized in practice (Antolin et al., 2019).
Remeshing and adaptation: No dynamic remeshing—the cross-tile parameter variations are enacted purely via control point shifts within the fixed trivariate grid.
Physics generalization: Existing demonstrations address mechanical, thermal, and mass transport phenomena; extending to coupled multi-physics, strongly nonlinear or biological behaviors remains an open direction (Antolin et al., 2019).
Optimization scale: For large discrete assembly spaces (e.g., $2^{27}$ possible vertex code assignments), combinatorial approaches (factorial search, row-graph path enumeration, integer-coded genetic algorithms) replace full enumeration (Doškář et al., 2023, Rocca et al., 2021, Rocca et al., 2021).

Notable avenues for extension include hierarchical multi-scale tiling, co-optimization of macro-mapping and micro-parameters, automated code selection for spatial inhomogeneity, direct coupling with tetrahedral meshing for FE analysis, and application to origami structures via Lagrangian isometric mapping constructions (including energy functional minimization for programmable deformation) (Liu et al., 2023).

7. Summary Table: DiT in Representative Domains

Domain	Tile Parameters	Optimization Approach
Porous microstructures	Geometry, topology, materials	Black-box/lumped FE, param. sweeps
Metamaterials (robotics)	Vertex-coded modules (Wang tiles)	Factorial search, combinatorial enum.
Phased arrays	Tile positions/topologies, array coefficients	Integer-coded GA, height-function enum.
GEMM accelerators	Hardware/software schedules, tiling parameters	Executable model + cost-based pruning
Origami/kirigami	Curves, folding angles, local thickness	Lagrangian ODE, variational opt

This framework supports a broad class of parametric, constraint-integrated, and performance-driven designs, offering a general methodology for the synthesis, analysis, and fabrication of both physical and abstract systems through the systematic optimization of tile-based assemblies (Antolin et al., 2019, Doškář et al., 2019, Doškář et al., 2023, Rocca et al., 2021, Rocca et al., 2021, Shen et al., 15 Dec 2025, Liu et al., 2023, Kaji et al., 2016, Schaad, 2015).