Computational Design and Optimization of Discrete Shell Structures Made of Equivalent Members

Abstract

This paper presents a computational design method for generating discrete shell structures using sets of equivalent discrete members. This study addresses the challenge of reducing the geometrical variety in discrete shell elements by introducing a method to design and optimize constituent members considering their similarity, approximation of the double-curved architectural surface, and buildability. First, we employed a relaxation-based computational form-finding method to generate a discrete topology with planar quad faces and an approximated smooth, double-curved surface. Then, we perform clustering and optimization based on face similarities concerning the minimization of deviations from the smooth surface approximation, and the dihedral angle between the planes of neighboring elements and their optimal intersection plane. The proposed approach can reduce the geometrical differences in discrete shell elements while satisfying the user-defined error threshold. We demonstrated the viability of our method on various structured topologies with different boundary conditions, support settings, and total face counts, while explicitly controlling inter-element facing angles for assembly ready contacts. This enables mold-based prefabrication with repeatable components.

1. Introduction

For centuries, shell structures have been used in architecture due to their high weight-to-span ratio and structural effectiveness [1]. These 3D surface-like shapes have a thickness much smaller than other dimensions, and their performance depends on the double curved surface geometry [2]. While digital design enables easy formal exploration, constructing double-curved surfaces poses significant challenges. In this regard, recent developments in computer-aided design have made double-curved surfaces more accessible. However, their complex geometry creates design, fabrication, and assembly challenges. Studies have focused on rationalization to simplify the construction of complex surfaces [3,4,5], with a key approach being surface discretization into smaller members. Thus, architectural rationalization has gained attention to design shell geometries that are practical and cost-effective to fabricate.

Discrete shells are geometric tiling arrangements on shell surfaces that utilize connected members without overlap [6]. The resultant assembly consists of uniquely shaped elements. Discretizing double-curved surfaces into equivalent elements requires a complex design study. These structures are characterized by polygonal meshes for numerical accuracy, with edges as beams and faces as panels. Various methods exist to reduce the number of unique edges in architectural structures [7,8] and sphere tessellations [9,10].

Architectural works with double-curved surfaces comprise uniquely shaped elements, often requiring customized fabrication [11], while design variables and objectives are influenced by manufacturing technologies and materials. Several discrete shell design studies utilize prefabrication techniques and simple joining methods, including timber plates [12], cross-laminated timber plates [13], and timber cassettes [14]. The Armadillo Vault demonstrates a compression-only discrete shell using shaped limestone voussoirs [15]. Concrete discrete shell examples include the ribbed concrete shell prototype [16] and a discrete shell with post-tensioned precast members [17].

Reducing unique elements simplifies CNC machining and 3D printing setups [18] and enables mold-based prefabrication through the reuse of molds. Using identical elements allows backup creation and reduces transportation costs if waste occurs [19]. One approach to reducing the geometric variety in double-curved architectural surfaces is modeling with discrete equivalence classes. Studies often simplify geometry using fixed topology with triangular or quadrilateral faces to explore equivalent elements through clustering and vertex optimization [20,21]. Alternatively, increasing formwork reusability is investigated to fabricate curved surface panels [22] or triangular folding elements [23], where congruent elements are manufactured and reshaped to different sizes. A recent study [24] presents a method to discretize double-curved freeform surfaces into congruent polyhedral members with gaps.

Finding the overall form is crucial in shell design. Several numerical form-finding methods exist, including the Force Density Method [25], Particle-spring Method [26], Dynamic Relaxation Method [27], and Thrust Network Analysis [28]. The initial step is finding a shape that satisfies the structural equilibrium; additional constraints for feasibility and buildability can be included through optimization [29,30]. Several studies focus on shape optimization for structural performance [31,32,33]. Others use geometry processing for self-supporting forms [34,35,36,37,38,39]. Also, artificial neural networks are employed to address complex structural design challenges [40], such as physics-informed neural networks [41] and mechanics-informed models [42]. Self-supporting structures are not our focus; however, integrating computational form-finding methods into the initial formal exploration process of shell geometry provides a plausible starting point for load-bearing shells.

The assembly of discrete shells remains complex, particularly when scaffolding systems are avoided [43,44]. A staged block insertion method using suspension cables reduces costs and ensures structural stability [45]. Robotic assembly can support specific block configurations [46]. Assembly sequence optimization depends on block count, discretization pattern, and shape, requiring equilibrium calculations and temporary support considerations [24]. The assembly process is outside the scope of this study and will be addressed separately.

The novelty of this work lies in combining relaxation-based form-finding, similarity-driven clustering, and fabrication-oriented optimization into a unified parametric workflow, specifically targeting mold-based prefabrication for discrete shells. Unlike [24,47], which address clustering or panelization separately, our method integrates surface approximation control, facing angle alignment, and cluster congruency optimization within a single optimization framework. In this study, we find the relaxed mesh and continuum surface approximation considering boundary conditions, supports, loads, and topological relationships. Planarized quad faces are grouped by geometrical similarities, calculated through vertex correspondences and alignment. An optimization procedure transforms faces within sets into congruent shapes while preserving topology. The 3D discrete shell elements are generated and optimized for buildability by minimizing dihedral angle deviations between neighboring elements. The proposed methodology provides architects and engineers with an adaptive computational tool that optimizes shell structures, considering both their architectural expression and fabrication efficiency. The method accounts for buildability by controlling the dihedral angles between contact faces and ensures geometric similarity within a user-defined error threshold. This study presents a novel approach to discrete shell design by integrating computational form generation with geometric simplification, while also incorporating geometry processing procedures that enable the use of efficient and scalable fabrication strategies.

2. Materials and Methods

In response to the complexities associated with constructing double-curved structures, this research addresses the challenge of reducing the number of unique elements in discrete shell structures. Distinct from existing methodologies that focus on structural performance or the generation of self-supporting forms, the present approach introduces an innovative computational workflow that integrates relaxation-based form-finding, surface approximation, similarity-based quad face clustering, and geometric optimization of discrete shell elements. A detailed flowchart of the proposed approach is presented in Figure 1. In this paper, the term face refers to a single quadrilateral face in the mesh, element refers to the 3D solid derived from a face, and panel is used in fabrication contexts to denote the 3D element derived from a face.

2.1. Computational Form-Finding

In the present study, a computational workflow is developed that includes computational form-finding based on an initial 2D topology definition. We used quad meshes to reduce the initial difference in the geometry of elements and to prevent the generation of irregular shapes, especially near the boundaries. The initial two-dimensional mesh $M_{2D}$ is defined by a set of vertices $V\in R^2$, edges $E$, and faces $F$. This mesh defines the topological and geometrical groundwork for subsequent computational operations:

$$M_{2D} = \left\{V,E,F\right\}, V=\left\{v_i\right\}, v_i=\left(x_i,y_i\right).$$

(1)

To achieve a structured quad-based tessellation suitable for computational form-finding, the initial mesh $M_{2D}$ was refined by subdividing existing mesh faces by introducing new vertices along edges and within faces. The mesh subdivision process is recursive, and in this discrete shell study, the subdivision level is a parameter that controls the size of the discrete members. Specifically, to create a uniform subdivision, an edge $e_{ij}$ connecting vertices $v_i$ and $v_j$ is subdivided by placing a new vertex $v_{ij}^{*}$ at the midpoint:

$$v_{ij}^{*}={\displaystyle \frac{1}{2}}\left(v_i+v_j\right),$$

(2)

Subsequently, each face $f_k$ of the original mesh $M_{2D}$ initially defined by vertices $v_{k1},v_{k2},...,v_{kn}$, is subdivided into smaller quadrilaterals by connecting these newly introduced midpoint vertices. This results in a refined quad-based mesh $M_{quad}$ defined as:

$$M_{quad} = \left\{V_{quad},E_{quad},F_{quad}\right\}, V_{quad}=\left\{v_i,v_{ij}^{*}\right\}.$$

(3)

Piker’s works inspired the computational form-finding approach employed to generate the geometry of shell structures [48]. The problem is formulated as a nonlinear optimization problem. Specifically, the objective is to determine a minimal energy configuration through iterative numerical methods, ensuring structural equilibrium under given boundary conditions and constraints. The refined mesh $M_{quad}$ serves as the basis for the three-dimensional form-finding process. The equilibrium shell geometry is then obtained by minimizing the total potential energy $\mathsf{\Pi }\left(V_{quad}\right)$, expressed as the sum of elastic (spring-like) potential energy and gravitational potential energy:

$$\mathsf{\Pi }\left(V_{quad}\right)={\displaystyle \frac{1}{2}}{\sum }_{\left(i,j\in E_{quad}\right)}{k}_{ij}{\left(‖v_i-v_j‖-l_{ij}\right)}^2+{\sum }_{i\in V_{quad}}{m}_{i}g{z}_i,$$

(4)

where $k_{ij}$ is the stiffness coefficient assigned to the edge $e_{ij}$, and $l_{ij}$ denotes the prescribed rest length. This formulation follows the standard total potential energy expression for spring–mass systems under gravity [27,48]. Deviations from these rest lengths induce internal stresses represented by the squared difference in current and rest lengths. The second summation accounts for the gravitational potential energy, where $m_i$ indicates the mass assigned to the vertex $v_i$, $g$ is gravitational acceleration, and $z_i$ is the vertical coordinate of the vertex $v_i$. The optimization problem requires the gradient of the energy function concerning vertex positions to vanish at equilibrium, yielding the condition:

$${\nabla }_{{\mathit{V}}_{\mathit{q}\mathit{u}\mathit{a}\mathit{d}}}\mathsf{\Pi }\left(V_{quad}\right)=0.$$

(5)

Expanding this equilibrium condition explicitly results in the following nonlinear equations:

$${\displaystyle \frac{\partial \Pi }{\partial v_i}}=\sum _{j\in N\left(i\right)}{k}_{ij}\left(1-{\displaystyle \frac{l_{ij}}{\Vert v_i-v_j\Vert }}\right)\left(v_i-v_j\right)+ m_{i}g{e}_z=0 ,$$

(6)

where $N\left(i\right)$ denotes the vertex adjacency set associated with the vertex $v_i$, and $e_z$ is the unit vector in the vertical (Z-Axis) direction. Numerical solutions are obtained through an iterative method, Dynamic Relaxation, wherein vertex positions are incrementally updated:

$$v_i^{\left(t+\Delta t\right)}=v_i^{\left(t\right)}+a\mathsf{\Delta }{t}^2{\displaystyle \frac{\partial \Pi }{\partial v_i}} ,$$

(7)

$$‖v_i^{\left(t+\Delta t\right)}-v_i^{\left(t\right)}‖<ϵ .$$

(8)

The parameter $a$ acts as a damping coefficient that controls the iterative process’s convergence stability and numerical efficiency. The vertex position update rule is adapted from the explicit Dynamic Relaxation method [27], where acceleration terms are replaced by force gradients. Convergence of the iterative procedure is monitored by evaluating vertex displacement between consecutive iterations, with equilibrium achieved upon satisfying the convergence threshold $ϵ$ which is defined as $1\times {10}^{-4}$ in this study. This value is a dimensionless convergence threshold, representing the normalized maximum vertex displacement between two successive iterations, expressed as a fraction of the average edge length of the mesh. The refined mesh $M_{quad}$ and the relaxed mesh are shown in Figure 2. The coincident mesh vertices on the red edges are characterized as fixed on the XY Plane. In contrast, the coincident vertices on the black boundary edges are only allowed to translate along the Z Axis.

Following the computational form-finding, a planarization step was performed to ensure all quad mesh faces are planar. This planarization process is formulated as shown in Equation (9), to minimize deviations from planarity for each quad face $f_{quad,i}$ defined by vertices $v_{i1},v_{i2},v_{i3},v_{i4}$.

$$\mathit{min}\sum _{f_{quad,i}}|\left(v_{i2}-v_{i1}\right)\times \left(v_{i4}-v_{i1}\right)·\left(v_{i3}-v_{i1}\right)| .$$

(9)

The resultant 3D relaxed mesh vertices $V_{quad}^{3D}$ were employed to construct a smooth, continuous surface using Rhino’s native Patch Surface method [49] that can be described as a parametric surface:

$$S\left(u,v\right)=\sum _{i=1}^n{N}_i\left(u,v\right)v_i^{3D}, u,v\in \left[\mathrm{0,1}\right] ,$$

(10)

where $N_i\left(u,v\right)$ represents basis functions defined by a spline-based interpolation method native to Rhino (i.e., Non-Uniform Rational B-Splines—NURBS). This approach is selected for its compatibility with the mesh topology and its relatively low computational cost. While other smooth-surface interpolation methods, such as subdivision surfaces, can provide higher-order continuity, they require additional management of boundary conditions and are more computationally demanding. In the context of this study, the chosen approach was found to offer a sufficient balance between fidelity and efficiency. The accuracy of this approximation is sensitive to vertex positioning, especially in areas of high curvature or non-uniform face distribution. This factor should be considered when applying the method to complex geometries. The vertex positions obtained from the relaxation process serve as control points for surface interpolation. The Patch Surface method seeks to minimize deviations between the interpolated surface $S(u,v)$ and the input vertices are formulated as follows:

$$\underset{S(u,v)}{\mathit{min}}\sum _{i=1}^n {‖S\left(u_i{v}_i\right)-v_i^{3D}‖}^2 .$$

(11)

2.2. Similarity-Based Clustering and Optimization

Following the generation of the form-found discrete shell geometry with planar faces and its smooth surface approximation, several problems need to be addressed regarding the generation of discrete shell members. A key step in this investigation is to evaluate how “close” two quad faces are regarding shape. Considering two quad faces, $P$ and $Q$ each represented by an ordered set of $n$ vertices, the aim is to find their best-fit alignment [21,47]. The similarity between them can be evaluated by first aligning faces $P$ onto $Q$ by utilizing rigid-body transformation (i.e., translations, rotations, and their reversed order when flipping is allowed) that minimizes the deviation between corresponding vertices. To determine the best-fit alignment for each pair, all possible vertex correspondences are analyzed with $Q$ fixed as shown in Figure 3.

Assuming the vertex arrangement of $Q$ remains unchanged, any vertex in $P$ may be selected as the initial vertex, yielding $n$ distinct permutations. For a given vertex correspondence, the best-fit mapping process can be explored using the approach described in [50]. Following the analysis of all the possible alignments, the similarity metric $d(P,Q)$ is defined as the average Euclidean distance between corresponding vertices after optimal alignment:

$$d\left(P,Q\right)={\mathit{min}}_{j=1}^{N_{\mathit{perm}}}{\left({\displaystyle \frac{1}{n}}\sum _{i=1}^n{‖T_{P_j\to Q} \left(P_{j,i}\right)-Q_i‖}^2\right)}^{{\displaystyle \frac{1}{2}}} ,$$

(12)

where $T_{P_j\to Q}$ is the best-fit rigid transformation for the $j$-th permutation of the vertices $P$, $P_{j,i}$ is the $i$-th vertex of $P$ under the $j$-th permutation, $Q_i$ is the $i$-th vertex of $Q$, and $N_{perm}$ equals $n$ (or $2n$ if both orientations are taken into account). This metric extends the rigid-body alignment error used in [50] by averaging vertex deviations over all permutations of correspondences, and ensures invariance to vertex labeling and orientation. This metric evaluates the average deviation per vertex, and lower values indicate higher geometric similarity; with $d(P,Q)=0$ the faces are congruent. This metric is the foundation for clustering faces into sets of nearly congruent shapes. After computing the shape similarity, the quad mesh faces are clustered using an iterative process. The next step is to partition the set of faces into clusters based on their shapes. Initially, a random face within each group is selected as an initial centroid. Each face is then best-fit aligned to its group centroid, and the centroid is updated by averaging the positions of corresponding vertices:

$$\overline{P_i}={\displaystyle \frac{1}{N}}\sum _{j=1}^N{P}_{j,i}, i=1,\dots ,n ,$$

(13)

where $N$ denotes the number of faces in the cluster and $P_{j,i}$ is the $i$-th vertex of the $j$-th face after alignment. After updating the centroid, the distance between each face and the centroid is determined by utilizing the similarity metric $d(P,P^{¯})$. If the clustering does not meet a predefined number of groups, the face with the maximum distance (i.e., the most dissimilar) from its centroid is selected to seed a new group. Faces are reassigned to the group with the smallest distance, and the centroids are recalculated iteratively. This farthest point sampling combined with a k-means-like update ensures that faces within each cluster share a high degree of similarity.

While clustering groups together similar faces, the intrinsic variability within a group may still be important for fabrication purposes. The objective of the optimization stage is to transform the faces within each cluster toward congruent shapes. This is achieved by enforcing the equivalence of key geometric features [21,47]. In this context, each quad face is characterized by the edge lengths $l_{ij}$ for the $j$-th edge of face $i$, the angle edges $l_{ij}^{\prime }$ a virtual edge associated with the turning angle at each vertex to find the angular relationship at each vertex, and signed distances $h_{ij}$ from the vertex $j$ of face $i$ to the best-fit plane. Within each cluster, the objective is to equalize these features across all faces. For a group of $F$ faces, the similarity term $F_s$ is formulated as:

$$F_s=\sum _{i=1}^F\sum _{j=1}^{n_i}\left[{\left(l_{ij}-{\overline{l}}_{ij}\right)}^2+{\left(l_{ij}^{\prime }-{\overline{l}}_{ij}^{\prime }\right)}^2+{\left(h_{ij}-{\overline{h}}_{ij}\right)}^2\right] ,$$

(14)

where ${\overline{l}}_{ij}$, ${\overline{l}}_{ij}^{\prime }$, ${\overline{h}}_{ij}$ represent the corresponding mean values: average edge length, angle edge length, and signed distance within the group, respectively. By minimizing $F_s$, the method forces the members of each group to approach a congruent form. For architectural purposes, it is often important that the adjusted faces remain close to the target double-curved surface. To achieve this, an additional term is included that penalizes deviations of vertices from prescribed positions. This approximation term $F_a$ is formulated as:

$$F_a={\omega }_{sur}\sum _{i=1}^V{d}_{si}^2+{\omega }_{bou}\sum _{j=1}^{V_{bou}}{d}_{bj}^2 ,$$

(15)

where $d_{si}$ is the distance from a clothed vertex to the target surface, and $d_{bj}$ is the distance from a boundary vertex to the designated boundary curve. The weights ${\omega }_{sur}$ and ${\omega }_{bou}$ allow prescribing the relative importance of each constraint. In this approach, vertices on the XY Plane are characterized as support vertices, and their movement on the Z Axis is restricted since they will be used to define the supports of the discrete shell structure. The classification of relative vertices is presented in Figure 4.

2.3. Creating Discrete Shell Elements

After the planar mesh faces are optimized and grouped into discrete equivalence classes, the next phase involves generating 3D discrete shell elements suitable for fabrication. Each optimized mesh face is transformed into a 3D discrete shell element by extending its geometry with a finite thickness (Figure 5a) and controlling the facing angles between neighboring elements (Figure 5b). The objective is to ensure that when these elements are assembled, their side faces contact one another, within fabrication error margins. Each optimized face, represented by a planar polygon $F$ with vertices $\{v_1,v_2,\dots ,v_n\}$ and unit Normal $N$. A discrete shell element $S$ is constructed by offsetting $F$ along $\mathrm{N}$ by a specific thickness $\tau$ as shown in Figure 5a. This approach ensures the relative faces of each discrete member are parallel. Formally, the top face of a discrete shell element $F^{+}$ is defined concerning the bottom face ($F$) as:

$$F^{+}={\left\{v_i+\tau N\right\}}_{i=1}^n ,$$

(16)

with side faces generated by connecting the corresponding vertices of $F$ and $F^{+}$. In an assembled discrete shell structure, seamless contact between adjacent faces is crucial within fabrication and construction margins. To quantify the quality of these contacts, we introduce the concept of the facing angle ${\theta }^{*}$, which is defined as the angle between the outward-pointing normals of the contacting side face of two neighboring elements. If $N_i$ and $N_j$ are the normals of the side faces from elements $S_i$ and $S_j$, then the facing angle is computed as:

$${\theta }^{*}=\mathit{arccos}\left({\displaystyle \frac{N_i·\left(-N_j\right)}{‖N_i‖‖N_j‖}}\right) .$$

(17)

A facing angle of zero indicates perfect alignment, whereas deviation from zero introduces potential gaps or overlaps. To achieve seamless contact, an additional constraint is introduced. In this regard, let $P_{ij}$ is the optimal intersection plane of two discrete shell elements $S_i$ and $S_j$. The planes that the elements’ contacting side faces can be defined as $P_i$ and $P_j$. Each of these planes induces dihedral angles ${\theta }_{ij}$ and ${\theta }_{ji}$ between $P_i$–$P_{ij}$, and $P_j$–$P_{ji}$, respectively; see Figure 5b. To prevent contact errors, an additional optimization is introduced that minimizes the diversity in facing angles across all contacts. Let $C$ denote the set of all pairs of contacting discrete shell elements. We define an evaluation metric $F_f$ for facing angle deviations:

$$F_f=\sum _{(i,j)\in C}{\omega }_{\mathit{ij}}\left|{\theta }^{*}-\left({\displaystyle \frac{{\theta }_{ij}+{\theta }_{ji}}{2}}\right)\right| ,$$

(18)

where $C$ is the set of all pairs of contacting elements ${\theta }^{*}$ is the target facing angle, and ${\omega }_{ij}$ are weights that may emphasize critical contacts. This quadratic formulation allows the use of gradient-based optimization methods (e.g., nonlinear conjugate gradient) to adjust the positions of vertices along the element boundaries.

The adjustments are constrained such that the underlying topology and the overall thickness $\tau$ are maintained, and the modified geometry remains as close as possible to the optimized planar face. In our framework, the overall objective function combines the previous shape similarity term $F_s$ and approximation to the smooth surface $F_a$ which was used to optimize the 2D mesh faces and the new facing angle term $F_f$ as:

$$F_t={\omega }_s{F}_s+{\omega }_a{F}_a+{\omega }_f{F}_f ,$$

(19)

where ${\omega }_s$ influences the intra-cluster shape congruence by controlling the similarity between set elements, ${\omega }_a$ controls the approximation of the discrete shell topology to the smooth surface geometry, and ${\omega }_f$ allows seamless contact for each pair of discrete shell elements. The optimization is performed by adjusting the positions of the mesh vertices while preserving the connectivity (topology) of the mesh in which the nonlinear conjugate gradient method [51] is used to solve the minimization of $F_t$. Minimization of $F_t$ allows the formation of discrete shell structures by using sets of discrete equivalence classes, each consisting of equivalent geometries.

3. Results

We actualize our methodological investigation in the Grasshopper–Rhino 7.38 (Robert McNeel & Associates, Seattle, WA, USA) environment on a desktop computer with a 5.0 GHz CPU and 64GB of memory. All numerical inputs (boundary conditions, face counts, support settings) are generated parametrically in Grasshopper. Data for the figures and tables are directly exported from the optimization output logs. Our approach involves computational form finding for a discrete shell structure, based on a 2D boundary mesh definition and quadrilateral discretization. This is followed by clustering the faces based on their similarity and optimizing the 3D discrete shell elements by minimizing the angle deviation of dihedral angles between contacting faces and the optimal intersection plane. Three error metrics are introduced to evaluate the efficiency of our approach, namely the similarity error ${\delta }_s$ for assessing the intra-cluster similarities by normalizing the largest volume of the element with the average volume in the relative cluster, the facing angle error ${\delta }_f$ to evaluate the largest deviation of elements’ contacting side faces from the relative optimal intersection plane, and the surface approximation error ${\delta }_a$ to calculate the maximum distance between the discretized mesh and the smooth surface approximation. The similarity error ${\delta }_s$ is expressed in cubic centimeters (${\mathrm{c}\mathrm{m}}^3$) since it is based on normalized volume differences. The facing angle error ${\delta }_f$ is expressed in radians ($\mathrm{r}\mathrm{a}\mathrm{d}$). The surface approximation error ${\delta }_a$ is expressed in meters ($\mathrm{m}$) as a maximum Euclidean distance. The optimization is concluded when ${\delta }_s$, ${\delta }_f$ and ${\delta }_a$ are reached by the user-defined thresholds considering architectural design requirements and fabrication error margins of 0.005, 0.035 and 0.05, respectively.

To evaluate the performance of our modeling approach, we tested two distinct cases on 2D topologies. As described comprehensively in [52], a key criterion in topology design is the number of singularities (vertices with irregular valency). Structured meshes with fewer singularities are preferred for identifying structural patterns, whereas unstructured meshes result in disorganized element flows. In this regard, Case 1 involved varying boundary conditions, number of singularities, support vertex settings, and mesh face counts to assess our approach’s adaptability to different geometrical settings. On the other hand, Case 2 focused on varying mesh face count under consistent boundary conditions and support vertex settings. We defined the thickness of discrete shell elements $\tau$ as 10 cm. The weights of the relative objectives regarding the similarity of the sets ${\omega }_s$, surface approximation of discretized mesh ${\omega }_a$, and facing angle deviation ${\omega }_f$ are defined as ${\omega }_s={\omega }_f=1 ,{\omega }_a=0.5$ since ${\omega }_a$ has a relatively higher influence on the convergence by restraining the similarity-based clustering process. The form-found discrete geometry and its double curved surface approximation are supplied to the optimization process. The optimization of each shape starts with the lowest possible target set number 1, and the set number $k$ is incrementally raised until the user-defined thresholds are satisfied. We characterized the minimum set number $k_{min}$ as the lowest number of sets of equivalent discrete members that satisfy the constraints of ${\delta }_s$, ${\delta }_f$ and ${\delta }_a$.

In the first part of our experiments, we tested the adaptability of our approach. We applied our method to 2D structured quad meshes with different topological settings in terms of number of singularities and number of faces, and different boundary conditions and support vertex settings as shown in Figure 5. In this regard, we employed a shape with no singularities and no boundary vertices (Figure 6a), a shape with no singularities (Figure 6b), a shape with only a single singularity (Figure 6c), and a shape with multiple singularities (Figure 6d). The resultant discrete shell alternatives are presented in Figure 7, and the numerical data of our implementation is outlined in

Table 1. Numerical data regarding Case 1. $N$ is the total number of faces in a 3D relaxed mesh, $l_{avg}$ is the average edge length, $k_{min}$ is the minimum number of sets that satisfies the user defined thresholds ${\delta }_s$ is the similarity error constraint, ${\delta }_f$ is the facing angle error constraint, ${\delta }_a$ is the surface approximation error constraint, and the runtime of each implementation is presented.

Figure No	$\mathit{N}$	${\mathit{l}}_{\mathit{a}\mathit{v}\mathit{g}} \left(\mathbf{m}\right)$	${\mathit{k}}_{\mathit{m}\mathit{i}\mathit{n}}$	${\mathit{\delta }}_{\mathit{s}} \left(\mathbf{c}{\mathbf{m}}^3\right)$	${\mathit{\delta }}_{\mathit{f}} \left(\mathbf{r}\mathbf{a}\mathbf{d}\right)$	${\mathit{\delta }}_{\mathit{a}} \left(\mathbf{m}\right)$	Runtime (min)
Figure 7a	912	0.864	7	0.002	0.033	0.010	14.5
Figure 7b	512	0.514	7	0.003	0.034	0.015	12.1
Figure 7c	384	0.485	10	0.004	0.035	0.008	24.8
Figure 7d	768	0.871	23	0.005	0.033	0.033	42.0

.

The results of Case 1 indicate that the surface curvature’s complexity significantly affects both the $k_{min}$ and runtime of the optimization. The resultant discrete shell alternatives in Figure 7a,b resulted in lower $k_{min}$ numbers compared to the other shapes while satisfying the error metrics due to their relatively less complex surface curvature characteristics. Also, the discrete shell alternatives with singularities (Figure 7c,d) resulted in higher $k_{min}$ values since the number of singularities significantly increases the geometrical differences in initial mesh faces. In addition, the distribution of discrete shells into relative sets depends on the geometrical variety in the faces of discretized geometries. The initial geometrical variety among the discrete faces caused by the singularities affects the element distribution within the sets. Therefore, the optimization of the shapes that included initial singularities in their geometries resulted in relatively small sets to satisfy the similarity threshold. The element distribution within the sets for each shape is outlined in

Table 2. The distribution of elements into the respective sets in Case 1. $k_{min}$ is the minimum number of sets that satisfies the user-defined thresholds, $n_{max}$ is the number of elements in the largest set, and $n_{min}$ is the number of elements in the smallest set.

Figure No	${\mathit{k}}_{\mathit{m}\mathit{i}\mathit{n}}$	${\mathit{n}}_{\mathit{m}\mathit{a}\mathit{x}}$	${\mathit{n}}_{\mathit{m}\mathit{i}\mathit{n}}$	Element Distribution
Figure 7a	7	152	76	152, 152, 152, 152 152, 76, 76
Figure 7b	7	128	12	128, 112, 76, 64, 60, 60, 12
Figure 7c	10	162	6	162, 66, 42, 42, 30, 12, 12, 6, 6, 6
Figure 7d	23	136	4	136, 80, 78, 72, 52, 44, 40, 32, 30, 30, 28, 24, 20, 20, 14, 12, 12, 12, 8, 8, 8, 4, 4

.

In the second part of our experiments, we employed similar structured meshes except for their face count to test the effect of the number of faces. To achieve this, we employed a 2D structured mesh and we applied a recursive subdivision procedure to generate alternatives with the same boundary conditions and support settings while only having different face counts. The resultant geometries of subdivision implementation are presented in Figure 8 with their resultant face counts. The subdivision process is guided by predefined edges to generate only quadrilateral mesh faces. Also, each shape has the same number of singularities. The optimized discrete shell alternatives for Case 2 are presented in Figure 8, and the numerical data of our implementation is outlined in

Table 3. Numerical data regarding Case 2. $N$ is the total number of faces in a 3D relaxed mesh, $l_{avg}$ is the average edge length, $k_{min}$ is the minimum number of sets that satisfies the user defined thresholds, ${\delta }_s$ is the similarity error constraint, ${\delta }_f$ is the facing angle error constraint, ${\delta }_a$ is the surface approximation error constraint, and the runtime of each implementation is presented.

Figure No	$\mathit{N}$	${\mathit{l}}_{\mathit{a}\mathit{v}\mathit{g}}\left(\mathbf{m}\right)$	${\mathit{k}}_{\mathit{m}\mathit{i}\mathit{n}}$	${\mathit{\delta }}_{\mathit{s}}\left(\mathbf{c}{\mathbf{m}}^3\right)$	${\mathit{\delta }}_{\mathit{f}}\left(\mathbf{r}\mathbf{a}\mathbf{d}\right)$	${\mathit{\delta }}_{\mathit{a}}\left(\mathbf{m}\right)$	Runtime (min)
Figure 9a	48	3.343	8	0.003	0.035	0.048	9.1
Figure 9b	192	1.715	10	0.004	0.034	0.032	13.5
Figure 9c	768	0.871	14	0.005	0.026	0.022	21.4
Figure 9d	3072	0.439	18	0.005	0.023	0.008	56.6

.

The resultant discrete topologies have support vertices and boundary vertices as well as singularities in their geometries while their smooth surface approximation has relatively complex curvature characteristics. Besides the surface complexity, we confirmed that the face count also directly affects the convergence time and error metrics. The optimization algorithm was able to find suitable design alternatives that satisfy the error metrics starting from $k_{min}=8$ for the shape with the lowest face count (Figure 9a) in comparison to the other alternatives. The number of sets that include equivalent elements has a proportional relationship to the total face count; the geometry with the largest face count of ($N=3072$) satisfied the error metrics with the largest $k_{min}$, comparatively. Also, the increase in the number of faces resulted in lower surface approximation and the maximum facing angle deviation error metrics while requiring more computation time to satisfy the similarity metric. In addition, we confirmed that the initial geometrical variety of the discrete geometry highly influences the distribution of elements in their respective sets based on similarity. The resultant element distribution is outlined in

Table 4. The distribution of elements into the respective sets in Case 2. $k_{min}$ is the minimum number of sets that satisfies the user defined thresholds, $n_{max}$ is the number of elements in the largest set, and $n_{min}$ is the number of elements in the smallest set.

Figure No	${\mathit{k}}_{\mathit{m}\mathit{i}\mathit{n}}$	${\mathit{n}}_{\mathit{m}\mathit{a}\mathit{x}}$	${\mathit{n}}_{\mathit{m}\mathit{i}\mathit{n}}$	Element Distribution
Figure 9a	8	8	4	8, 8, 8, 8, 4, 4, 4, 4
Figure 9b	10	42	6	42, 36, 24, 20, 18, 16, 12, 10, 8, 6
Figure 9c	14	106	12	106, 100, 90, 86, 56, 52, 52, 52, 48, 36, 34, 24, 20, 12
Figure 9d	18	728	10	728, 492, 434, 292, 260, 188, 128, 108, 104, 82, 76, 54, 40, 40, 14, 12, 10, 10

.

Figure 10, Figure 11 and Figure 12 present the convergence histories for the error metrics regarding the similarity error ${\delta }_s$, the facing angle error ${\delta }_f$, and the surface approximation error ${\delta }_a$ in relation to the number of sets $k$. The performance of these metrics was evaluated across multiple mesh densities. These plots address both the optimization performance and the influence of mesh refinement on the results. In the plots, the curves show a steep initial decrease in error values during the early iterations, followed by a gradual stabilization, indicating consistent convergence behavior. Coarser meshes exhibit higher starting errors and more substantial early reductions, whereas finer meshes begin with lower errors and converge more gradually, reflecting a smaller margin for improvement. This pattern highlights a fundamental trade-off: coarse meshes offer greater computational efficiency but may fail to capture important geometric and structural details. In contrast, finer meshes provide higher fidelity at the cost of longer processing times. In the context of shell structures, these observations are structurally relevant since mesh size directly affects the accuracy of curvature representation, the identification of potential stress concentrations, and the precision of assembly tolerances. The presented convergence plots thus provide both a quantitative record of optimization progress and a practical reference for selecting mesh densities that balance geometric accuracy, computational cost, and fabrication constraints.

The runtime results in Table 1 and Table 3 reflect the direct influence of mesh density on computational costs. Higher mesh resolutions lead to longer runtimes due to the quadratic growth of pairwise similarity calculations during clustering. Despite this, convergence plots in Figure 10, Figure 11 and Figure 12 indicate that the majority of optimization progress occurs within the first 30–40% of iterations, after which improvements taper off. This trend suggests that early termination strategies could be implemented to improve computational efficiency without significantly compromising solution quality. Furthermore, the algorithm generated a nearly uniform distribution of elements within the sets when the input discrete topologies had no singularities and their double-curved surface approximations had relatively simple surface curvatures. In contrast, optimizing geometries with singularities resulted in significant deviations in the number of elements in their respective sets, as the algorithm prioritizes the separation of significantly different geometries into their respective sets. In addition, the geometries that possess inherent symmetries are employed in this study. It was observed that, although there is no integration of symmetry recognition procedures in the workflow, the approach was efficient enough to create nearly symmetrical discrete shell results, as shown in Figure 13.

4. Discussion

This study addresses the prominent rationalization challenge of utilizing geometries with discrete equivalence classes in architectural discrete shell design problems. Unlike existing methodologies, our approach focuses on minimizing the geometrical variety of discrete shell elements through a computational workflow that integrates relaxation-based form-finding, similarity-based clustering, and geometric optimization of discrete shell elements, considering user-defined thresholds for architectural appearance and fabrication constraints. The proposed method departs from prior work through three primary innovations. First, rather than operating solely at the surface/panel level, we generate 3D elements with explicit thickness and assess contact geometry between neighboring elements. Second, we introduce a facing angle objective (Equations (16)–(18)) that prioritizes assembly ready interfaces within prescribed tolerances. Third, the optimization jointly enforces cluster congruency (Equation (14)) and surface fidelity (Equation (15)) together with contact alignment in a single objective (Equation (19)), targeting mold reuse and repeatable fabrication. This combination is specifically oriented to prefabrication logistics (mold count reduction and tolerance management), which extends beyond the primary goals of prior clustering/panelization frameworks.

We applied quadrilateral discretization to the geometries to reduce the difference in the initial shape of elements and to prevent the generation of irregular shapes, especially near the boundaries. Our findings confirmed the hypothesis that employing planar quadrilateral meshes simplifies the initial geometric complexity and effectively reduces the shape variety in discrete members. The results demonstrate that discretization using structured quadrilateral meshes generated by the computational form-finding process allows the creation of element groups with minimal variation. As the comparative tests demonstrate, meshes without singularities and with relatively simple boundary conditions require fewer discrete equivalence classes and converge faster, aligning with prior studies that emphasize the benefits of structured meshing in reducing geometric complexity. The face count of the discrete meshes directly affects the complexity and efficiency of the optimization process. The results show a clear correlation between increased mesh density, improved double-curved surface approximation, and reduced facing angle deviation in the expanse of increased computation time. Also, an interesting aspect of the research was the observed optimization accuracy to preserve inherent symmetries in optimized discrete shell design alternatives. Despite complex geometric configurations, the optimization process consistently resulted in near-symmetrical outputs, which contributed to further reducing the set numbers. This implicit recognition of symmetry is a byproduct of the clustering procedure rather than a targeted optimization objective. Integrating explicit symmetry metrics or symmetry-aware optimization methods could further enhance repetition rates and mold reusability, and is identified as a promising direction for future development.

In practical terms, the cluster count produced by our method directly specifies the mold count required for prefabrication. At the same time, the per-cluster element lists define production batches and reuse schedules. The facing angle metric serves as a tolerance proxy for assembly that indicates whether dry-fit contact is feasible or whether shimming/adjustable joints are required. The surface approximation metric informs finishing and machining allowances. Furthermore, the optimized geometries can be directly exported from Rhino as surface or solid meshes and imported into standard FEA software in STEP or IGES format. For discrete element analysis, each panel is exported as an individual solid with defined contact interfaces. This allows structural engineers to perform FEA on the same discretization and, if necessary, request targeted mesh refinement in critical regions.

Several limitations arise from our study that necessitate further exploration. Although clustering and optimization effectively reduce geometric variety, inherent complexities associated with structures featuring singularities or complex boundaries remain. Such cases increase computational time and the minimum number of discrete equivalence sets. While the method has been successfully applied to meshes with up to 3072 faces, scalability remains a challenge, as the computational cost increases non-linearly with mesh size. Large or highly irregular meshes require longer runtimes and more memory resources due to the similarity assessment step, which involves calculating pairwise distances between elements. The quality of the initial mesh is also a factor in achieving stable convergence since poorly shaped or irregularly proportioned meshes may slow optimization and lead to less efficient clustering outcomes. Future research could focus on advanced clustering algorithms or adaptive discretization strategies to handle such complexities more efficiently by creating fewer sets with a uniform distribution of elements. Furthermore, the approach depends on user-defined thresholds for similarity tolerances, which are case-specific. While this flexibility enables adaptation to various fabrication contexts, it also means that the results are sensitive to these input parameters. In this regard, the presented convergence histories (Figure 10, Figure 11 and Figure 12) and runtime data (Table 1 and Table 3) offer guidance for selecting practical mesh densities and stopping criteria.

The facing angle optimization evaluates the planarity of the transverse faces of the neighboring discrete shell elements. However, minor curvature or fabrication deviations could cause local misalignments in practice. This limitation could be mitigated by tolerance-based clustering or adjustable joint interfaces. Also, the practical challenges of assembling discrete shell structures without temporary scaffolding or support elements remain unresolved within the scope of our current study. Although we have optimized geometric alignment and minimized facing angles, physical assembly feasibility and structural stability during assembly require further in-depth investigation, which opens up interesting directions for research that include integrated robotic fabrication techniques or automated assembly sequencing strategies.

The similarity metric in Equation (12) is defined in terms of Euclidean vertex-to-vertex distances between corresponding mesh faces. This choice ensures computational efficiency and maintains a consistent, material-independent basis for geometric comparison, which is essential for evaluating element congruency across varied mesh topologies. While this approach effectively captures geometric alignment, it does not directly account for fabrication-related factors such as angular deviations between face normals, variations in element thickness, or tolerance margins that may arise during manufacturing and assembly. Therefore, the introduced error metrics incorporate user-defined thresholds, which serve as case-specific specifications of allowable geometric deviation. These thresholds can be adjusted according to material thickness and fabrication precision requirements. Furthermore, flipping of mesh faces is not permitted during similarity-based optimization to ensure that surface normal directions remain consistent across all elements, preserving correct orientation for assembly. While the current framework allows for the adjustment of user-defined thresholds to reflect material thickness and tolerance requirements, future work could expand this capability to infer threshold values from material and process constraints automatically. In addition, embedding material anisotropy constraints, such as grain direction in timber or formwork-induced shrinkage in concrete, would allow the workflow to account for directional performance and deformation risks. Maintaining consistent surface normal orientation is already enforced in the current workflow to avoid geometric inconsistencies during assembly, and this constraint will remain critical in material-specific extensions.

While explicit structural analysis and physical prototyping are beyond the scope of the present work, several aspects of the proposed workflow inherently support structural plausibility and fabrication readiness. The form-finding stage produces geometries in static equilibrium, providing an efficient starting point for shell action. Low dihedral-angle deviations and a close approximation of the smooth surface suggest that the optimized forms can maintain efficient force flow under typical shell loading conditions. However, their self-supporting behavior may require further investigation and adjustments once integrated with structural analysis workflows. Although the mesh size here is chosen based on geometric rationalization goals, finer meshes can better capture stress gradients in structural analysis. Our method can accommodate varying mesh densities; however, the trade-off between fabrication simplification and structural accuracy requires further study. Due to minor contact errors, the relative structural performance assessment can be challenging using analysis methods such as the rigid-block equilibrium methods [31,32,44]. Therefore, further studies could explore combined optimization frameworks incorporating structural performance criteria to enhance practical applicability. In addition, the proposed method can effectively reduce the geometrical variety in discrete shell members and allows the utilization of mold-based fabrication techniques. Testing the structural behavior and suitability of a specific material can be a direction for another study. Lastly, the fabrication of shell elements and physical experimentation could provide understanding to improve the proposed discrete shell design method.

5. Conclusions

This study addresses the prominent rationalization challenge of utilizing geometries with discrete equivalence classes in architectural discrete shell design problems. We formulated a clustering-based optimization procedure to generate sets of equivalent discrete shell members corresponding to a form-found discretized shell topology and its double-curved architectural surface approximation. We characterized several metrics to assess the similarity of elements within each set, evaluate the relationship between planar quad mesh faces and smooth surface approximation, and their buildability by minimizing the dihedral angle between shell elements’ contacting side faces and their optimal intersection plane. We tested our proposed algorithm by employing various 2D structured meshes with different characteristics, including boundary conditions, singularity and support definitions, and quad face counts. Our approach effectively allows the creation of discrete shell design alternatives based on sets of equivalent shell members corresponding to the decision variables and user-defined constraints. Across all tested cases, the proposed method reduced the number of unique element geometries by 68–92% relative to the initial mesh, while maintaining surface approximation errors below 0.05 m and facing angle deviations below 0.035 rad.

The methodological workflow consisted of relaxation-based computational form-finding, ensuring equilibrium-based geometries, followed by quad mesh planarization to generate a viable baseline for clustering. Subsequently, similarity-based clustering and optimization stages were executed to categorize and refine mesh faces into minimal groups of geometrically congruent elements. The final optimization stage addressed a crucial fabrication constraint by controlling the dihedral angles between neighboring shell elements, ensuring seamless assembly within defined fabrication tolerances. This optimization strategy successfully balanced the objectives of geometric approximation accuracy and element congruency.

We validated our approach through computational experimentation across multiple scenarios characterized by differing boundary conditions, support vertex settings, face counts, and singularities in discrete mesh topologies. The experiments showed the methodology’s adaptability, successfully optimizing geometries from simpler meshes without singularities to more complex configurations with multiple irregularities. Notably, structured quadrilateral discretization allowed the reduction in the geometric diversity, which is crucial in practical architectural applications that utilize prefabrication and mold-based manufacturing techniques. The quad-based discretization was proven to be especially effective, as it inherently minimizes geometric variance and supports symmetrical clustering patterns. The computational experiments highlighted the inherent capability of the proposed optimization methodology to recognize symmetry. This feature significantly reduced the number of unique prefabricated molds required, further minimizing manufacturing complexity and cost.

Overall, the proposed method contributes to computational design studies in architecture by providing a viable strategy for rationalizing and simplifying discrete shell structures. This study presents an innovative computational design approach to address the complexities associated with discrete shell structures, with a primary aim of significantly reducing the number of unique geometric elements. The results clearly demonstrate the method’s effectiveness in managing geometric complexity and systematically minimizing variations among discrete shell elements, which not only contributes to more practical and economical fabrication processes but also allows the utilization of mold-based prefabrication techniques in the discrete shell design problems.