An Integrated Shape Optimization Method for Hybrid Structure Consisting of Branch and Free-Form Surface

Featured Application

The integrated shape optimization method for branch-supported free-form surface structures proposed in this paper can provide an integrated shape optimization framework for other types of hybrid structures.

Abstract

Branching and free-form structures are widely used in large-span buildings. Their shapes are the main factors that affect their mechanical performances. Many studies have been carried out on the morphology of single structural systems, but less on hybrid structures. However, both of them often appear in the same building. In order to reflect the cooperative bearing of substructures in the optimization process of branch-supported free-form surface structures, this paper proposes a holistic shape optimization method. The proposed method extracts design variables based on the structural modeling process, and uses the coordinates in the parametric domain to realize a mathematical description of the positional relationship between substructures. Then, the sensitivity analysis method is used to adjust the position of design variables to reduce the overall strain energy, realizing the integrated shape optimization of this hybrid structure. The effectiveness of the method is validated through several numerical examples. The results show that the overall stiffness of the optimized structure has been significantly improved, and the process of integrated optimization is more convenient. Furthermore, the way of adjusting design variables directly affects the shape and mechanical performance of the optimized structure. This feature serves as a valuable design tool that can provide multiple feasible solutions for architectural and structural design.

1. Introduction

With rapid progress in architectural design and construction technology, architecture is developing towards complexity, diversity, and novelty. Shape -resistant spatial structures, whose mechanical behaviors are closely related to architectural performance, are highly favored by engineers. As representatives of such structures, free-form and branching structures are widely used in modern large-span architecture [1,2,3,4,5,6]. There are also branch-supported free-form surface buildings that incorporate the geometric characteristics of these two structural forms [7,8].

The free-form surface structure has a flexible shape that is difficult to express with a single or a few analytical functions. Additionally, its geometry plays a crucial role in conveying the architectural intent as well as determining the structural mechanical performance [9,10]. A branching structure, also known as a tree-like structure, is a multi-level branching and three-dimensional expanding bar system. It is often used as a building support structure, which can effectively reduce the span of the roof without affecting the function of the lower space [11]. The shape and supporting position are important considerations in its design process [12]. The issue of creating a reasonable form for structures has garnered significant attention from scholars.

In terms of creating a reasonable form of free-form surface structure, scholars have developed numerous effective shape generation methods based on the principle of inverted hanging models and the theory of structural optimization. Before the widespread application of computers, the inverted hanging model experiment was the main means of determining the reasonable form of the free-form surface structure [2,13,14,15,16]. This method used flexible materials that could only withstand tensile forces to simulate the actual structural stress state and obtain a stable equilibrium shape under specified design conditions. Representative cases include the Deitingen motorway BP service station in Switzerland [17], the Heimberg Tennis Hall (also located in Switzerland) [18], and the Mannheim Multihalle in Germany [19]. While the concept behind the model experiment method is clear, the implementation process can be cumbersome when comparing multiple design options and may be affected by many uncertain factors during the operation [1,10,20]. As computer performance improves and numerical theories develop, numerical methods are gradually becoming the mainstream approach in the field of morphological generation, covering form-finding and structural optimization techniques. Form-finding methods mainly include the geometric stiffness method [21,22], the dynamic equilibrium method [23], and the stiffness matrix method [24,25,26]. Structural optimization is a systematic process that utilizes optimization algorithms to drive design variables and obtain an optimal structural shape, which is reflected by the objective function [1,20]. Common objective functions include minimizing structural weight [27], uniforming stress state [28,29], and minimizing strain energy (maximizing stiffness) [30,31,32,33]. Studies have shown that minimizing structural strain energy can effectively reduce structural bending moments, resulting in structures with higher structural efficiency [10,28,29]. Compared to form-finding, structural optimization has a wider range of applications [1,20].

In terms of creating a reasonable form of branching structure, its development process is similar to that of free-form surface structure. Early scholars also sought a reasonable form of branching structure based on the principle of model experiments, relying on axial force transmission to bear loads, mainly including the inverse hanging model, thread model, etc. [4,6,11,14,15,16], which were successfully applied to the design process of the Riyadh government center, the Stuttgart Airport, and other architectural projects [34]. In the field of the numerical creation of reasonable forms for branching structure, scholars also focus on two directions: form-finding and structural optimization techniques. Typical form-finding methods include the graphic statics [35], the inverted hanging recursive method [12,36], the force density method [37], the element-clustered method [38,39], the quasi-mechanism method [40], and others. The methods based on optimization techniques include bi-directional evolutionary structural optimization (BESO) [41], extended evolutionary structural optimization (EESO) [42], sensitivity analysis [43], and others.

Most of the existing studies on morphological generation focused on free-form surfaces or branching structures separately. However, a branch-supported free-form surface building is a rigid spatial structure composed of shell and beam elements, and its mechanical performance is the comprehensive manifestation of the synergistic effect of both the upper and lower parts of the structure. Previous literature has shown that, when optimizing the stiffness of a structure, the stiffness of non-optimized regions significantly affects the optimization results [44]. In contrast to the conventional approach of separately optimizing individual substructures in a hybrid structural shape optimization process, this paper proposes an integrated shape optimization method for branch-supported free-form surface structures with the goal of improving the overall performance. This method effectively captures and incorporates the collaborative load-bearing effects among its substructures during the optimization process. The remainder of this study is as follows. The initial structural modeling method suitable for the optimization process based on the geometric features of the hybrid structure is proposed in Section 2. Then, the integrated shape optimization method aimed at minimizing the overall strain energy is established in Section 3. In Section 4, several numerical examples are given to verify the effectiveness of the method. Finally, the conclusions are presented in Section 5.

2. Initial Structural Model

The branch-supported free-form surface structure is comprised of two distinct parts with noticeable geometric differences: the upper free-form surface structure and the lower tree-like supporting structure. It belongs to the shell-beam structural system, where the shape of the free-form surface, the pattern of branches, and the location of intersection points are important indicators that reflect the geometric characteristics of the hybrid structure. To accurately capture the intricate and diverse features of the upper free-form surface, we employ B-spline surface interpolation to define its shape based on given data points. Regarding the lower tree-like supporting structure, its shape can be represented by the coordinates of branch nodes and their topological relationships. The relative positioning of the two components will be described by the parameter values of intersection points on the upper surface.

2.1. The Free-Form Surface Model Based on the B-Spline Technique

The B-spline technique is a commonly used method of representing free-form curves and surfaces in the field of computer-aided geometric design (CAGD). The fundamentals of B-spline surfaces in [45] are summarized here. A B-spline surface S(u, v) is a piecewise polynomial surface obtained by taking a bidirectional net of control points, two knot vectors, and the products of the univariate B-spline functions:

$$S\left(u,v\right)={\displaystyle \sum _{i=0}^n{\displaystyle \sum _{j=0}^m{N}_{i,p}\left(u\right)N_{j,q}\left(v\right)P_{i,j}}}$$

(1)

where the {P_i,j} (i = 0, 1, …, n; j = 0, 1, …, m) is a (n + 1) × (m + 1) matrix, and form a bidirectional control net, the p and q are the degrees in the u and v directions, the {N_i,p(u)} and {N_j,q(v)} are the B-spline basis functions defined on the knot vectors U = {u₀, u₁, …, u_n+p+1} and V = {v₀, v₁, …, v_m+q+1}, the {N_i,p(u)} can be calculated by the Equation (2). The formula for calculating {N_j,q(v)} has the same form.

$$\begin{array}{l}{N}_{i,0}\left(u\right)=\left\{\begin{array}{l}1 \mathrm{if} u_i\le u<u_{i+1}\\ 0\hspace{1em}\mathrm{otherwise}\end{array}\right.\\ N_{i,p}\left(u\right)=\frac{u-u_i}{u_{i+p}-u_i}{N}_{i,p-1}\left(u\right)+\frac{u_{i+p+1}-u}{u_{i+p+1}-u_{i+1}}{N}_{i+1,p-1}\left(u\right)\end{array}$$

(2)

Equation (1) can be used to compute the point on a B-spline surface at fixed parameter values (u, v). In this case, the design variables of the free-form surface structure are the control points. Utilizing the strong convex hull property and local modification scheme of B-spline surfaces to estimate their positions can approximate the structural shape envisioned by the designer. However, accurately controlling specific points on the free-form surface remains a challenge. To address this issue, we propose using the B-spline surface interpolation technique.

Assuming that given a set of (n + 1) × (m + 1) data points {P_k,l} (k = 0, 1, …, n, l = 0, 1, …, m), we use the equally spaced method to assign a set of parameter values (u_k, v_l) for each point P_k,l for convenience, as shown in Equation (3):

$${\overline{u}}_k=k,{\overline{v}}_l=l\left(k=0,1,\dots ,n;l=0,1,\dots ,m\right)$$

(3)

In order to reflect the distribution of (u_k, v_l), we use the averaging technique of Equation (4) to define the knot vector U. Likewise, the same method can be applied to define the knot vector V.

$$\begin{array}{l}{u}_0=\cdots =u_p=0,u_{n+1}=\cdots =u_{n+p+1}=n\\ u_{i+p}=\frac{1}{2}\left({\overline{u}}_{i-1}+{\overline{u}}_{i+p}\right)=i+\frac{1}{2}\left(p-1\right),i=1,2,\cdots ,n-p\end{array}$$

(4)

Further, we can set up the system of linear equations as shown in Equation (5).

$${\displaystyle \sum _{i=0}^n{\displaystyle \sum _{j=0}^m{N}_{i,p}\left({\overline{u}}_k\right)N_{j,q}\left({\overline{v}}_l\right)P_{i,j}={\overline{P}}_{k,l}}}$$

(5)

By solving Equation (5), we can determine the coordinates of the unknown control points {P_i,j}. Thus, using Equation (1), the free-form surface interpolating data points {P_k,l} are obtained.

Figure 1 illustrates an example of a B-spline surface interpolant. The surface is a (2, 2)th-degree B-spline surface interpolating the set of 6 × 6 data points, and the uniformly divided parameter domain mesh is mapped into Euclidean three-dimensional space to obtain a discrete, relatively uniform free-form surface mesh.

2.2. The Geometric Representation of the Branching Support Structure Model

For a branch-supported free-form surface structure, both the main trunk and each level branch are composed of beam elements. In the finite element method, the geometry of the beam element is defined by nodal coordinates and their topological relationships. Considering that the research in this paper revolves around the overall performance of hybrid structures, we classify the branch nodes into two types based on their positions and label them accordingly. The first type refers to the intersection nodes between the branches and free-form curved roof, and their parameter values on the surface are directly stored in the geometric information base to reflect their positions relative to the upper surface. The second type refers to the non-intersection nodes of the branching structure, and their location is saved using Cartesian coordinates.

Table 1. A branching support structure model and its geometric information.

Structure	Node Number	Type *	Coordinate	Link Number	Link
	1	NXPT	(−8.0, −8.0, 0.0)	1	(1, 2)
	2	NXPT	(−8.3, −8.3, 3.9)	2	(2, 3)
	3	NXPT	(−9.9, −8.4, 7.1)	3	(2, 4)
	4	NXPT	(−8.4, −9.9, 7.0)	4	(2, 5)
	5	NXPT	(−7.5, −7.5, 7.9)	5	(3, 6)
	6	XPT	(0.97, 1.36)	6	(3, 7)
	7	XPT	(1.05, 1.65)	7	(3, 8)
	8	XPT	(1.23, 1.45)	8	(4, 9)
	9	XPT	(1.36, 0.97)	9	(4, 10)
	10	XPT	(1.45, 1.23)	10	(4, 11)
	11	XPT	(1.65, 1.05)	11	(5, 12)
	12	XPT	(1.52, 1.52)	12	(5, 13)
	13	XPT	(1.58, 1.79)	13	(5, 14)
	14	XPT	(1.79, 1.58)

* XPT represents the intersection node, and NXPT represents the non-intersection node.

shows a branching support structure model and its geometric information.

2.3. The Composition of Design Variables for Branch-Supported Free-Form Structure

According to the modeling process of the branch-supported free-form structure in Section 2.1 and Section 2.2, the design variables can be composed of the following three parts (Figure 2):

(1)

The physical coordinates P^S of the data points of the free-form surface are defined as P^S = {P^S_{i, j}}, where P^S_{i, j} = {x^S_{i, j}, y^S_{i, j}, z^S_{i, j}}^T. Adjusting P^S in physical space can alter the shape of the surface.

(2)

The parameter values P^O of the intersection points between the branches and free-form curved roof are defined as P^O = {${\mathsf{P}}_i^{\mathrm{O}}$}, where ${\mathsf{P}}_i^{\mathrm{O}}$ = {$u_i^{\mathrm{O}}$, $v_i^{\mathrm{O}}$}^T. Adjusting P^O in parametric space can change the supporting location of the branching structure relative to the upper surface.

(3)

The physical coordinates P^B of the non-intersection points of the branching structure is defined as P^B = {${\mathsf{P}}_i^{\mathrm{B}}$}, where ${\mathsf{P}}_i^{\mathrm{B}}$ = {$x_i^{\mathrm{B}}$, $y_i^{\mathrm{B}}$, $z_i^{\mathrm{B}}$}^T. Adjusting P^B in physical space can alter the shape of the branching structure.

3. The Integrated Shape Optimization Method for Branch-Supported Free-Form Surface Structure

The establishment of a mathematical model of the optimization problem of branch-supported free-form surface structure is shown in Equation (6):

$$\left\{\begin{array}{l}find:P=\left(P^{\mathrm{S}},P^{\mathrm{O}},P^{\mathrm{B}}\right)\\ \min .:f\left(P\right)=C\left(P\right)\\ s.t.:P\subseteq {\Omega }_0\\ \hspace{1em}\hspace{1em}g\left(P\right)\le 0\\ \hspace{1em}\hspace{1em}h\left(P\right)=0\end{array}\right.$$

(6)

where P are the optimization variables, which reflect the structural morphological characteristics, consisting of curved surface shape variables P^S, intersection point position variables P^O, and non-intersection point position variables P^B. In this paper, the objective function f(P) is the structural overall strain energy C(P), which is minimized through iterative solutions using sensitivity analysis. Ω₀ is the design allowable space, and g(P) and h(P) are the inequality and equality constraint conditions of the optimization model, respectively.

3.1. Establishment of the Shape Optimization Method for Branch-Supported Free-Form Surface Structure

The Taylor expansion of the structural strain energy C(P) at design variables P is given by:

$$C\left(P+\Delta P\right)=C\left(P\right)+{\displaystyle \sum _{i=1}^n\Delta P_i\frac{\partial C\left(P\right)}{\partial P_i}}+\frac{1}{2}{\left({\displaystyle \sum _{i=1}^n\Delta P_i\frac{\partial }{\partial P_i}}\right)}^{2}C\left(P\right)+R_2$$

(7)

where design variables P = (P₁, P₂, …, P_n)^T, ΔP represents the variation in design variables P, and R₂ is the Taylor series remainder term.

Let

$$\Delta P_i=-\lambda \cdot \frac{\partial C\left(P\right)}{\partial P_i}$$

(8)

When λ is a sufficiently small positive real number, substituting Equation (8) into Equation (7) and neglecting higher-order terms, we obtain:

$$C\left(P+\Delta P\right)\doteq C\left(P\right)-{\displaystyle \sum _{i=1}^n\lambda \cdot {\left(\frac{\partial C\left(P\right)}{\partial P_i}\right)}^2}$$

(9)

Equation (9) shows that the structural strain energy can be reduced if the design variables are slightly adjusted in the negative direction of the strain energy gradient, i.e.,

$$C\left(P+\Delta P\right)\le C\left(P\right)$$

(10)

By adjusting the design variables according to Equation (11) until the strain sensitivity approaches zero (i.e., ∂C(P)/∂P_i = 0 (i = 1, 2, …, n)). If further calculations are performed, the design variables will oscillate slowly around the extremum point, and the overall shape of the structure will remain almost unchanged. A reasonable structure with strain energy at a local minimum is obtained.

$$P_i^{k+1}=P_i^k-\lambda \cdot \frac{\partial C\left(P^k\right)}{\partial P_i^k}$$

(11)

where $P_i^k$ and $P_i^{k+1}$ represent the i-th design variable of the structure for the k-th and (k + 1)-th iterations, respectively, and λ represents the iteration step size.

From Equation (11), it can be observed that the variation of the design variable is jointly determined by the iteration step size and the strain sensitivity. The iteration step size is an empirical parameter. For the branch-supported free-form surface structure, due to the different dimensionalities of the three types of design variables, the same iteration step size may result in differences in the order of magnitude of the adjusted physical distances. In order to ensure the validity of Equation (10) (i.e., the variations should be sufficiently small in magnitude), we propose to set different gradient reference values, denoted as V_R = ($V_{\mathrm{R}}^{\mathrm{S}}$, $V_{\mathrm{R}}^{\mathrm{O}}$, $V_{\mathrm{R}}^{\mathrm{B}}$), for different types of design variables. The size of V_R is determined as follows:

In terms of the data points of the free-form surface:

$$V_{\mathrm{R}}^{\mathrm{S}}=\max \left(\left|\frac{\partial C}{\partial P_i^{\mathrm{S}}}\right|\right)$$

(12)

In terms of the intersection points between the branches and free-form curved roof:

$$V_{\mathrm{R}}^{\mathrm{O}}=\max \left(\left|\frac{\partial s}{\partial P_i^{\mathrm{O}}}\cdot \frac{\partial C}{\partial P_i^{\mathrm{O}}}\right|\right)$$

(13)

where, the expression “∂s/∂${\mathsf{P}}_i^{\mathrm{O}}$” represents the partial derivative of arc length “s” with respect to ${\mathsf{P}}_i^{\mathrm{O}}$.

In terms of the non-intersection points of the branching structure:

$$V_{\mathrm{R}}^{\mathrm{B}}=\max \left(\left|\frac{\partial C}{\partial P_i^{\mathrm{B}}}\right|\right)$$

(14)

Furthermore, introduce a sufficiently small positive real number ε to determine λ:

$$\lambda =\frac{\epsilon }{V_{\mathrm{R}}}$$

(15)

where, ε represents the maximum physical spatial variation of the design variables along their respective tuning directions allowed based on experience.

3.2. The Sensitivity of Branch-Supported Free-Form Surface Structure

Assuming that the loads F applied to the nodes of branch-supported free-form surface structure remain unchanged during the adjustment of the design variables (i.e., the loads are independent of the design variables), the partial derivative of the strain energy C with respect to the design variable P can be determined using the chain rule as given in Equation (16):

$$\frac{\partial C}{\partial P_i}={\displaystyle \sum _j\frac{\partial C}{\partial X_j}\cdot \frac{\partial X_j}{\partial P_i}}$$

(16)

where X_j represents the j-th node in the structural finite element model, P_i represents the i-th design variable in the structural geometric model, the expression “∂C/∂X_j” represents the partial derivative of the strain energy C with respect to X_j, and “∂X_j/∂P_i” represents the partial derivative of X_j with respect to P_i.

3.2.1. The Partial Derivative of Strain Energy C with Respect to Structural Nodes X

According to the finite element stiffness equation, we have:

$$KU=F$$

(17)

where K is the global stiffness matrix of the structure, U is the column vector of nodal displacements, and F is the column vector of nodal loads.

Differentiating both sides of Equation (17) with respect to X_i yields:

$$\frac{\partial U}{\partial X_i}=-K^{-1}\frac{\partial K}{\partial X_i}U$$

(18)

The expression for the strain energy C can be given as:

$$C=\frac{1}{2}{F}^{\mathrm{T}}\cdot U$$

(19)

Differentiating both sides of Equation (19) with respect to X_i yields:

$$\frac{\partial C}{\partial X_i}=\frac{1}{2}{F}^{\mathrm{T}}\cdot \frac{\partial U}{\partial X_i}$$

(20)

Substituting Equations (17) and (18) into Equation (20), we obtain:

$$\frac{\partial C}{\partial X_i}=-\frac{1}{2}{U}^{\mathrm{T}}\frac{\partial K}{\partial X_i}U$$

(21)

3.2.2. The Partial Derivative of Structural Nodes X with Respect to Design Variables P

According to Section 2.3, the design variables P of branch-supported free-form surface structure consist of three parts: the data points of surface (P^S), the intersection points between branches and surface (P^O), and the non-intersection points of branches (P^B). Similarly, the nodes X in the structural finite element model can also be divided into three types based on their positions: the nodes exclusively belonging to shell elements (X^S), the intersection nodes between shells and beams (X^O), and the nodes exclusively belonging to beam elements (X^B), i.e., X = {X^S, X^O, X^B}^T. The partial derivative ∂X/∂P can be calculated based on the influence area of the design variables P.

• For the design variable ${\mathsf{P}}_{j,k}^{\mathrm{S}}$ ∈ {P^S}, moving ${\mathsf{P}}_{j,k}^{\mathrm{S}}$ only affects the surface shape, i.e., {X^S, X^O}. The partial derivative ∂X_i/∂${\mathsf{P}}_{j,k}^{\mathrm{S}}$ can be computed as follows:

If X_i ∈ {X^S, X^O}, with its corresponding parameter value being (u_Xi, v_Xi), according to B-spline theory:

$$X_i={\displaystyle \sum _{r=0}^R{\displaystyle \sum _{s=0}^S{N}_{r,p}\left(u_{X_i}\right)N_{s,q}\left(v_{X_i}\right)P_{r,s}^{\mathrm{C}}}}$$

(22)

where, P^C is the set of control points on the B-spline surface. In this paper, P^C is determined through Equation (23):

$${\displaystyle \sum _{r=0}^R{\displaystyle \sum _{s=0}^S{N}_{r,p}\left({\overline{u}}_m\right)N_{s,q}\left({\overline{v}}_n\right)P_{r,s}^{\mathrm{C}}}}=P_{m,n}^{\mathrm{S}}$$

(23)

Differentiating both sides of Equations (22) and (23) with respect to the design variable ${\mathsf{P}}_{j,k}^{\mathrm{S}}$ yields:

$$\frac{\partial X_i}{\partial P_{j,k}^{\mathrm{S}}}={\displaystyle \sum _{r=0}^R{\displaystyle \sum _{s=0}^S{N}_{r,p}\left(u_{X_i}\right)N_{s,q}\left(v_{X_i}\right)\frac{\partial P_{r,s}^{\mathrm{C}}}{\partial P_{j,k}^{\mathrm{S}}}}}$$

(24)

$${\displaystyle \sum _{r=0}^R{\displaystyle \sum _{s=0}^S{N}_{r,p}\left({\overline{u}}_m\right)N_{s,q}\left({\overline{v}}_n\right)\frac{\partial P_{r,s}^{\mathrm{C}}}{\partial P_{j,k}^{\mathrm{S}}}}}=\frac{\partial P_{m,n}^{\mathrm{S}}}{\partial P_{j,k}^{\mathrm{S}}}={\delta }_{mj}\cdot {\delta }_{nk}$$

(25)

where δ_mj and δ_nk are Kronecker functions, δ_mj · δ_mj = 1 if and only if m = j, n = k, and 0 in other cases. The values of ∂X_i/∂${\mathsf{P}}_{j,k}^{\mathrm{S}}$ can be obtained by substituting the calculation result of Equation (25) into Equation (24).

If X_i ∈ {X^B}, $\frac{\partial X_i}{\partial P_{j,k}^{\mathrm{S}}}=0$.

• For the design variable ${\mathsf{P}}_j^{\mathrm{O}}$ ∈ {P^O}, assuming that its physical coordinate, obtained through Equation (22), is stored in the finite element model as node X_m, any movement of ${\mathsf{P}}_j^{\mathrm{O}}$ only affects X_m. The partial derivative ∂X_i/∂${\mathsf{P}}_j^{\mathrm{O}}$ can be computed as follows:

If i = m,

$$\frac{\partial X_i}{\partial P_j^{\mathrm{O}}}=\left[\begin{array}{l}\frac{\partial X_i}{\partial u_{P_j}}\\ \frac{\partial X_i}{\partial v_{P_j}}\end{array}\right]=\left[\begin{array}{l}{\displaystyle \sum _{r=0}^R{\displaystyle \sum _{s=0}^S{N^{\prime }}_{r,p}\left(u_{P_j}\right)N_{s,q}\left(v_{P_j}\right)P_{r,s}^{\mathrm{C}}}}\\ {\displaystyle \sum _{r=0}^R{\displaystyle \sum _{s=0}^S{N}_{r,p}\left(u_{P_j}\right){N^{\prime }}_{s,q}\left(v_{P_j}\right)P_{r,s}^{\mathrm{C}}}}\end{array}\right]$$

(26)

where, the derivative of a basis function is given by Equation (27), the detailed derivation process can be seen in [45]:

$${N^{\prime }}_{r,p}=\frac{p}{u_{r+p}-u_r}{N}_{r,p-1}\left(u\right)-\frac{p}{u_{r+p+1}-u_{r+1}}{N}_{r+1,p-1}\left(u\right)$$

(27)

Otherwise, $\frac{\partial X_i}{\partial P_j^{\mathrm{O}}}=0$.

• For the design variable ${\mathsf{P}}_j^{\mathrm{B}}$ ∈ {P^B}, assuming that its coordinate is directly stored in the finite element model as node X_m, any movement of ${\mathsf{P}}_j^{\mathrm{B}}$ only affects X_m. The partial derivative ∂X_i/∂${\mathsf{P}}_j^{\mathrm{B}}$ can be computed as follows:

If i = m, $\frac{\partial X_i}{\partial P_j^{\mathrm{B}}}=I$.

Otherwise, $\frac{\partial X_i}{\partial P_j^{\mathrm{B}}}=0$.

3.3. The Procedure of the Shape Optimization Method for Branch-Supported Free-Form Surface Structure

The operating procedure of the shape optimization method for branch-supported free-form surface structures can be summarized as shown in Figure 3:

(1)

Build the initial geometric model according to the architectural intention and constraints;

(2)

Classify the design variables into three types based on their influence areas;

(3)

Calculate the strain energy sensitivity of each design variable according to Section 2.2;

(4)

Determine the step size for each type of design variable using Equation (15).

(5)

Adjust the design variables using Equation (11).

(6)

Repeat steps (3) to (6) until the convergence condition is achieved.

(7)

The convergence condition is:

$$\left|C_k-C_{k-1}\right|\to 0 \mathrm{or} \max \left(V_{\mathrm{R}}\right)\to 0$$

(28)

4. Numerical Examples and Discussions

In this section, a numerical example is presented to verify the proposed method. The necessity of prioritizing the enhancement of overall structural performance as the objective in the optimization process of branch-supported free-form surface structure is illustrated through the discussion. Finally, the effectiveness of the optimization is demonstrated through a case study of a multi-tree-supported free-form surface structure.

Figure 4 shows an initial model of a branch-supported free-form surface structure. The structure spans a rectangular plane measuring 50.0 m × 50.0 m. The upper structure is represented by a (2, 2)th-degree B-spline surface that interpolates a set of 6 × 6 data points, and the lower structure is composed of the truck and two-level branching structure.

The upper shell thickness t = 0.1 m, elastic modulus E_C = 3.0 × 10⁴ MPa, Poisson ratio μ_C = 0.2, the lower tree-like structure members using solid circular section rod, the diameter of the 0th-level branch D₀ = 1.0 m, and the diameters of other level branches are scaled in a ratio of 0.7 from bottom to top, elastic modulus E_B = 2.1 × 105 MPa, Poisson ratio μ_B = 0.3. The vertical downward uniform load q = 3.0 kN/m² acts on the curved surface, and the nodes of the tree-like supporting structure are fixed.

The optimization scheme of the structure is usually determined according to the architectural design requirements. In this example, it is assumed that all three types of nodes forming the structure need to be adjusted, as shown in Figure 5a (Scheme I). The value points of the curved surface are only adjusted vertically (in the z direction), the intersection points between the tree structure and the free-form surface are adjusted within the surface, and the nodes of the tree structure except the intersection points are adjusted on the xoy plane where the point is located, while the bottom supports are fixed without adjustment. With a small positive value ε = 0.05, Figure 5b shows the shape when the structure optimization tends to be stable under this adjustment scheme.

In order to investigate the tension and compression distribution of the upper shell roof before and after optimization, the dimensionless index of Equation (29) is used to represent:

$${\lambda }_{\sigma }=\frac{{\sigma }_1^{\mathrm{mid}}}{{\sigma }_1^{\mathrm{mid}}+\left|{\sigma }_3^{\mathrm{mid}}\right|}$$

(29)

where ${\sigma }_1^{\mathrm{mid}}$ is the first principal stress of the shell, usually the tensile stress, and ${\sigma }_3^{\mathrm{mid}}$ is the third principal stress of the shell, usually the compressive stress. As can be seen from Equation (29), λ_σ ∈ [0, 1], when this index is close to 0, it is considered that the part is mainly under pressure; when this index is close to 1, it is considered that the part is mainly under tension. For the lower tree support structure, the tensile and compressive conditions can be measured by the local axial force FN.

Figure 6a–d, respectively, show the comparison between the tensile and compressive stress distribution of the upper freeform surface and the axial force distribution of the lower tree-like structure before and after optimization. It can be seen from Figure 6a that under the load set by this example, the upper freeform surface of the initial structure is mainly subjected to the film compressive stress, but the proportion of the film tensile stress in the local area, especially the area near the support, is larger. For the lower tree-like structure (Figure 6b), the local axial forces are all pressure, indicating that the tree-like support generation method proposed in Section 2.2 can meet the requirement that loads converge and transfer to supports step by step along the branches under the premise of reasonable setting of support areas, but the axial forces of different branches at the same level are greatly different. After optimization, the upper surface is uplifted as a whole in terms of visual effect, but the adjustments in different areas are slightly different. Corresponding to the tension and compression stress distribution of the surface, as shown in Figure 6c, the adjustment of the shape value points and the intersection points of the shell rods not only strengthens the overall arch effect but also strengthens the catenary effect of the local tension area. Due to the change in branch angle caused by the adjustment of nodes, the local axial force of the branches in the same order tends to be uniform, and the load distribution of the members is more reasonable.

To compare and analyze the influence of different design variable adjustment schemes on optimization results, this paper includes design variable adjustment schemes II–IV as shown in

Table 2. Changes in structural mechanical properties during evolution and the comparison of evolutionary results in different adjustment schemes.

No.	I			II			III			IV
Scheme *	PS	PO	PB	PS	PO	PB	PS	PO	PB	PS	PO	PB
	●	●	●	●	○	○	○	○	●	○	●	●
Ci/C0
$C_i^{\mathrm{S}}$/$C_0^{\mathrm{S}}$ $C_i^{\mathrm{B}}$/$C_0^{\mathrm{B}}$
$C_i^{\mathrm{S}\_\mathrm{m}}$/$C_i^{\mathrm{S}}$ $C_i^{\mathrm{B}\_\mathrm{m}}$/$C_i^{\mathrm{B}}$
Final Shape

* In this example, ${\mathsf{P}}_i^{\mathrm{S}}$ = {z_i}^T, ${\mathsf{P}}_j^{\mathrm{O}}$ = {u_j, v_j}^T, ${\mathsf{P}}_k^{\mathrm{B}}$ = {x_k, y_k}^T; ● adjustable, ○ non-adjustable.

. Except for the type of nodes to be adjusted, all other conditions (initial structure, load, boundary conditions, and material parameters) are the same as in Scheme I. Three groups of performance indicators are tracked to understand the changes in strain energy and the proportion of each component during the optimization process:

(1)

C_i/C₀: Strain energy change rate, measuring the change in structural strain energy;

(2)

${\mathrm{C}}_{\mathrm{i}}^{\mathrm{S}}$/${\mathrm{C}}_0^{\mathrm{S}}$ and ${\mathrm{C}}_{\mathrm{i}}^{\mathrm{B}}$/${\mathrm{C}}_0^{\mathrm{B}}$: Strain energy change rate in the shell and tree-like structures respectively, measuring the change in strain energy of each component in the hybrid structure;

(3)

${\mathrm{C}}_{\mathrm{i}}^{\mathrm{S}\_\mathrm{m}}$/${\mathrm{C}}_{\mathrm{i}}^{\mathrm{S}}$ and ${\mathrm{C}}_{\mathrm{i}}^{\mathrm{B}\_\mathrm{m}}$/${\mathrm{C}}_{\mathrm{i}}^{\mathrm{B}}$: Measures the proportion of membrane strain energy and axial force-induced strain energy in the i-th step of the free-form surface shell and tree-like structure, respectively.

where C₀ is the initial strain energy of the structure, C_i is the strain energy of the i-th step structure, ${\mathrm{C}}_0^{\mathrm{S}}$ is the initial strain energy of the free-form surface shell, ${\mathrm{C}}_{\mathrm{i}}^{\mathrm{S}}$ and ${\mathrm{C}}_{\mathrm{i}}^{\mathrm{S}\_\mathrm{m}}$ are the strain energy and membrane strain energy, respectively, of the i-th step free-form surface shell, ${\mathrm{C}}_0^{\mathrm{B}}$ is the initial strain energy of the tree-like structure, and ${\mathrm{C}}_{\mathrm{i}}^{\mathrm{B}}$ and ${\mathrm{C}}_{\mathrm{i}}^{\mathrm{B}\_\mathrm{m}}$ are the strain energy and axial force-induced strain energy, respectively, of the i-th step tree-like structure.

From the strain energy change rate and component proportion curves corresponding to each scheme in Table 2, it can be seen that the structural strain energy decreases significantly during the optimization process; the decrease rate shows a trend of gradually converging from fast to slow, and the proportion of membrane strain energy and axial force-induced strain energy in the structure also increases significantly, indicating that the morphological optimization method used in this paper promotes the evolution of the structure towards the shape that can resist external loads with membrane force and axial force, effectively improving the structural stiffness, and has good convergence.

In terms of different adjustment schemes, Scheme III (adjusting only the tree-like structure nodes) has the fastest convergence speed, reaching a good state in the initial optimization stage, indicating that the initial shape generated according to the tree-like structure generation method in Section 2.2 is more reasonable. From the perspective of strain energy change, Scheme I, which simultaneously adjusts all three types of nodes, has the best effect on improving structural stiffness; Scheme II adjusts only the surface-type value points and causes changes in the vertical position of the support points when changing the shape of the upper shell, so it also has some impact on the stiffness of the lower tree-like structure while improving the stiffness of the shell; Scheme III only adjusts the tree-like structure nodes (except for those intersecting with the surface), which can improve the stiffness of the lower tree-like structure and improve the support stiffness of the upper structure, so the strain energy of the shell will also change; Scheme IV simultaneously adjusts the tree-like structure nodes and the intersection points with the shell, significantly improving the stiffness of the lower tree-like structure, and making the support position more reasonable, so the strain energy of the upper structure will decrease, and the degree of decrease is higher than that of Scheme III.

The reduction in strain energy for the four different schemes was compared, and it was found that the optimization results varied depending on the design variable adjustment scheme employed. The more comprehensive the consideration of shape control factors (design variables), the more significant the improvement in structural stiffness (Scheme I > Scheme II, Scheme IV > Scheme III). This is mainly because, for the tree-like hybrid structure, the upper shell and lower tree-like structure work together to support the loads from the top, exhibiting clear synergistic effects. Adjustments made to a specific design variable can lead to changes in the local shape of the structure, which in turn affect the overall performance of the structure. Therefore, it is crucial to pursue holistic performance enhancement in morphological optimization. The integrated morphological optimization method adopted in this study takes into account the synergistic load-bearing effects of hybrid structures. While meeting the requirements of architecture and structural design, different adjustment schemes can also provide references for architectural design.

5. Conclusions

In this paper, an integrated shape optimization method for branch-supported free-form surface structure is proposed, which is suitable for the preliminary exploration stage of shape schemes in architectural design. The innovation points and conclusions of this study are summarized as follows:

(1)

Based on the theories of B-splines and the rules of branches, the geometric representation of freeform surfaces and tree-like structures has been achieved. The connection between the two is established through parameter space, providing convenience for the preliminary design of buildings.

(2)

Compared with the initial structure, the optimized structure relies more on membrane forces and axial forces to transmit loads. For tree-like structures, the distribution of axial forces within the same-level branches becomes more uniform, indicating a more reasonable load distribution of the branching system and effectively avoiding inefficient components. The overall strain energy of the structure decreases significantly, and the stiffness is effectively improved, leading to greatly improved load-bearing performance.

(3)

The morphological optimization method, aiming to minimize the overall strain energy of the structure, can fully consider the collaborative load-bearing effect between the upper and lower parts of the tree-like freeform structure. By comparing the optimization results under different adjustment schemes, the more adjustable shape control factors (design variables) there are, the more significant the optimization effect on structural stiffness. Under the premise of meeting the requirements of architecture and structure, different adjustment schemes can provide more references for architectural design.