Shape Optimization of Frame Structures Through a Hybrid Two-Dimensional Analytical and Three-Dimensional Numerical Approach ^†

Abstract

In this work, we propose a method for the shape optimization of frame structures using a mixed analytical–numerical approach. The goal is to achieve a uniform-strength frame structure, ensuring optimal material utilization and weight minimization. The optimization is performed in two calculation steps. The first step uses an analytical model based on the Timoshenko beam theory, where appropriate mathematical steps give a uniform-strength shape of the entire structure. Depending on the type of cross-section analyzed, the exact uniform-strength profile of each element is derived by solving for three parameters related to the forces and moments acting on the element. These parameters are obtained by solving a nonlinear system of equations, which includes the external and internal kinematic constraints of the structure, as well as equilibrium equations for each element. However, the solution obtained using the one-dimensional theory is limited in areas affected by boundary effects, such as the interconnection regions between elements and those near the supports, for a decay distance at least equal to the characteristic diameter of the section. To address this limitation, the second optimization step involves incorporating solutions that account for a triaxial stress field. This is typically carried out by discretizing the structure using the finite element method. The frame geometry obtained from the previous analytical solution is constructed, and the regions affected by boundary effects are optimized using the Biological Growth Method (BGM). This is an iterative, bio-inspired method modeled on the growth of trees, which increases trunk diameter in proportion to the loads experienced. The method is applied simultaneously to all regions where three-dimensional effects are significant, with the aim of achieving uniform strength in areas influenced by boundary effects. An important aspect of applying the BGM is maintaining the topology of the initial mesh, which is ensured through the use of mesh morphing techniques. The results of the two-step optimization process are shown on simple geometries involving few elements, and on more complex geometries of mechanical interest.

1. Introduction

Frame structures are among the most popular and versatile engineering solutions in mechanical and building design. These systems, composed of interconnected beams, offer an optimal combination of strength and rigidity, making them suitable for a wide range of applications in a variety of engineering fields [1,2,3]. The principle behind frame structures is to distribute loads efficiently by exploiting the geometry and rigid connection of structural elements.

In recent decades, with the advent of new materials and advanced analysis methods, frame structures have seen a significant evolution, improving both in terms of performance and sustainability. In particular, the development of additive manufacturing techniques [4] and homogenization algorithms [5] has made it possible to greatly extend the scope of application of these structures on ever-smaller scales.

On the basis of the above, it is evident that the optimization of frame structures is a very topical and pressing problem. Various approaches to the optimization of frame structures can be followed:

• Topological optimization [6], which focuses on the optimal distribution of material within a predefined space. The objective is to determine which portion of material must be retained or removed to achieve a structure that maximizes strength, rigidity or efficiency while minimizing weight. This approach is not limited to a predefined shape of the structure, but seeks to identify the best configuration of material within the available volume.
• Layout optimization [7], which focuses on the optimal positioning of beams within the frame structure. In this case, the objective is to find the best configuration of members, i.e., nodes’ location, that minimizes stresses while maintaining the desired structural performance.
• Shape optimization [8,9], which focuses on optimizing the shape of individual beams generally to reduce weight or stress peaks.

In this work, an innovative optimization workflow is proposed to combine analytical and numerical methods for the optimization of frame structures. The workflow idea is illustrated in Figure 1.

In the first optimization step, an analytical approach is adopted: starting from the Timoshenko beam theory, the uniform strength condition is imposed to compute the optimal beam cross-section. This method makes it possible to obtain a first optimized geometry in a very short time. However, the main limitation is the failure of Timoshenko beam theory to consider three-dimensional effects, making the analytical solution less effective in the presence of striction or joint zones where a triaxial stress state appears. To take edge effects into account, finite element analysis must be introduced.

To this end, an FEM model and a numerical method capable of optimizing the most critical areas is required. The Biomechanical Growth Method (BGM) was chosen because it is compatible with the analytical approach described above, as it allows the introduction of a target stress, is automatable and is very robust. Therefore, the second optimization step is based on a numerical method that allows the critical areas to be resolved, also taking into account three-dimensional effects. Applying the BGM to the already-optimized geometry allows the optimal solution to be found with significantly fewer iterations.

Combining numerical and analytical methods is a highly innovative approach, because it successfully merges accuracy with workflow efficiency. The approaches referenced in the literature typically focus exclusively on either numerical or analytical methods. The critical challenge lies in transferring information from the low-fidelity analytical model to the high-fidelity numerical model.

In this work, an automated method based on mesh morphing is proposed to address this issue. This approach offers multiple advantages. Compared to purely numerical approaches, starting from an already-optimized configuration allows for identifying the optimum with fewer analyses. Compared to purely analytical methods, it can also handle more complex geometries, accounting for three-dimensional stress effects, resulting in more precise and reliable outcomes.

The applications of this method are extensive. In this thesis, two applications are proposed: the optimization of a frame structure and a lattice-like structure. In general, the method can be extended to any mechanical component, and is particularly relevant given the current development of metamaterials with lattice-like structures.

2. Theoretical Background

This paragraph outlines the theoretical foundations underlying the workflow applied in this study.

2.1. Uniform-Strength Shape from Timoshenko Beam Model

Many structural optimization algorithms, whether gradient-based or heuristic, share a significant limitation: the need to repeatedly perform finite element analysis (FEA) at each evaluation of the objective function, i.e., at every iteration. To substantially accelerate the optimization process, we propose an approach that first obtains an optimized configuration from a 1D analytical model based on beam theory. This allows the subsequent 3D optimization, using the BGM, to start from an already-optimal state and focus on refining the solution in only a few calculation steps. These refinements are applied selectively, targeting areas where they are most needed—specifically, regions of constriction and junctions between elements.

The detailed analytical procedure followed is described in [8,9], and the key concepts are summarized below.

As a reference, consider the beam shown in Figure 2, with length $L$, variable height $h\left(x\right)$, and subject to forces and moments $\left(F_1,\dots ,F_6\right)$ at the end-nodes.

The six forces and moments must fulfill the equilibrium equations:

$$F_4=-F_1$$

(1)

$$F_5=-F_2$$

(2)

$$F_6=F_2 L-F_3$$

(3)

As results from the planar Timoshenko beam model [10,11], the stress state is biaxial, with the approximations for the axial and mean shear stresses as follows:

$${\sigma }_x\left(x,y\right)={\displaystyle \frac{N\left(x\right)}{A}}-{\displaystyle \frac{M\left(x\right)}{I}}y$$

(4)

$${\tau }_x={\displaystyle \frac{T\left(x\right)}{A}}$$

(5)

where the axial and shear forces $N,T$ and the bending moment $M$ are linked with the applied forces, as follows:

$$N\left(x\right)=-F_1$$

(6)

$$T\left(x\right)=-F_2$$

(7)

$$M\left(x\right)=F_2 x-F_3$$

(8)

The shape h(x) that ensures the iso-stress condition can be obtained from Equation (4) by setting ${\sigma }_x\left(x,y=\pm h\left(x\right)/2\right)={\sigma }_{iso}=constant$, obtaining the following:

$${\sigma }_{iso}={\displaystyle \frac{\left|N\left(x\right)\right|}{A\left(x\right)}}+{\displaystyle \frac{\left|M\left(x\right)\right|}{I\left(x\right)}}{\displaystyle \frac{h\left(x\right)}{2}}$$

(9)

where ${\sigma }_{iso}$ is the uniform-strength (iso-resistance) stress, and $A\left(x\right)$ and $I\left(x\right)$ are the variable area and moment of inertia of the cross-section.

Equation (9) considers only the axial stress, i.e., does not take into account the shear stress. This is not a problem, since according to the Jourawsky’s approximation [10,12], shear stress vanishes at the top and bottom of a beam cross-section. The information of the mean shear in Equation (5) is used to set a minimum thickness of the cross-section in the region in which the bending moment is null, avoiding the absence of material.

Considering a circular cross-section of radius R(x), inserting into Equation (9) the expression of A(x), I(x) as functions of the radius, and solving R(x) accordingly, one obtains the following:

$$R\left(x\right)=\sqrt[3]{-{\displaystyle \frac{2\left|M\left(x\right)\right|}{\pi {\sigma }_{iso}}}} \left[ \sqrt[3]{-1+\sqrt{\Delta \left(x\right)}}+\sqrt[3]{-1-\sqrt{\Delta \left(x\right)}} \right]$$

(10)

where

$$\Delta \left(x\right)=1-{\displaystyle \frac{1}{108 \pi {\sigma }_{iso}}} {\displaystyle \frac{{\left|N\left(x\right)\right|}^3}{M^2\left(x\right)}}$$

(11)

It is worth pointing out that Equation (10) is simplified a lot if the axial force is neglected, which is a recurring case in the frame of lattice structures, where stress due to the bending moment is dominant.

Equation (10) is the analytical expression of the beam shape that ensure the iso-stress condition. It can be used to integrate, analytically or numerically, the kinematic of the beams (elastic line), to obtain the six nodal displacements and rotations $\left(u_1,\dots ,u_6\right)$ (Figure 2) that are functions of the nodal forces of the elements. The analytical equations of the kinematics can be used to form a nonlinear system of equations for the functions of the unknown nodal forces $\left(F_1,\dots ,F_N\right)$, in which, at each node, the conditions of equilibrium, constraint and kinematic congruence are imposed. The solution of the nonlinear system is obtained by means of the trust-region algorithm. For more details about the numerical implementation, the reader can be refer to [9].

2.2. RBF Mesh Morphing

Mesh morphing is a widely utilized technique in engineering for various applications. The core concept involves defining source points with known displacements and interpolating these displacements across the mesh nodes. This approach is commonly employed in fluid–structure interaction [13], where the fluid mesh is deformed based on structural displacements, as well as in optimization tasks [14], where mesh-level parameters are adjusted, and in automated optimization processes [15]. One key advantage of mesh morphing is that it allows for working with a consistent mesh throughout the process. Radial basis functions (RBFs) are particularly well suited for mesh morphing, as they facilitate the interpolation of displacements imposed onto the mesh nodes at source points. At each node, the displacement applied is proportionally to its distance from the surrounding source points. From a mathematical point of view, it is necessary to solve the following linear system [16]:

$$f^x\left(x\right)={\sum }_{i=1}^m{\gamma }_i^x\varphi \left(‖c_i-x‖\right)+{\beta }_1^x+{\beta }_2^x{x}_1+{\beta }_3^x{x}_2+{\beta }_4^x{x}_3$$

(12)

$$f^y\left(x\right)={\sum }_{i=1}^m{\gamma }_i^y\varphi \left(‖c_i-x‖\right)+{\beta }_1^y+{\beta }_2^y{x}_1+{\beta }_3^y{x}_2+{\beta }_4^y{x}_3$$

(13)

$$f^z\left(x\right)={\sum }_{i=1}^m{\gamma }_i^z\varphi \left(‖c_i-x‖\right)+{\beta }_1^z+{\beta }_2^z{x}_1+{\beta }_3^z{x}_2+{\beta }_4^z{x}_3$$

(14)

where $\gamma$ are the weights and ϕ are the radial basis functions; the polynomial term has a stabilizing function. The system is solved by applying, as boundary conditions, the value of the known displacement at the source points and the orthogonality of the coefficients. There is a wide range of RBFs available. The interpolation behavior, while guaranteeing the value imposed on the center points, depends on the specific RBF chosen. The computational cost and approaches to RBF solutions also vary depending on the type of function.

2.3. Biological Growth Method (BGM)

The BGM is a stress-driven approach inspired by the adaptive behaviors of biological structures, aimed at optimizing structural components. Proposed by Mattheck and Burkhardt in 1990 [17] (Figure 3), and recently extended by Porziani et al. [18,19], the BGM is based on the observation that natural systems, such as bones and tree trunks, adjust their shapes in response to external loads by adding material in regions of high stress, and removing it in low-stress areas. This process results in an optimized geometry with a uniform von Mises stress distribution on free surfaces.

The growth mechanism in the BGM follows a linear relationship between surface stress and material adaptation, governed by a threshold stress value (σ_tℎ). Material is added or removed based on local stress levels, with the magnitude and direction of the displacement determined by the BGM stress data and limited by a maximum displacement. The modification of the surface shape is achieved through an offset technique, where node displacements are applied in the direction of the surface normal:

$$S_{node}={\displaystyle \frac{{\mathsf{\sigma }}_{node}-{\mathsf{\sigma }}_{th}}{{\mathsf{\sigma }}_{max}-{\mathsf{\sigma }}_{min}}}d$$

(15)

where $S_{node}$ is the displacement applied in each node, ${\mathsf{\sigma }}_{node}$ is the node stress, ${\mathsf{\sigma }}_{th}$ is the reference stress, ${\mathsf{\sigma }}_{max}$, ${\mathsf{\sigma }}_{min}$ are the maximum and minimum values of the stress, respectively, and $d$ is a scale parameter.

Automatic optimization integrates BGM stress data, using mesh morphing tools to adjust the structure. This allows for efficient, data-driven modifications that align with the stress distribution, resulting in improved structural performance.

2.4. Performance Factors

To evaluate the effectiveness of the proposed method, several factors were introduced to compare the geometries obtained at various optimization steps. Specifically, two factors were introduced that assess the stress distribution on the surface (factor $f_1$) and throughout the entire volume (factor $f_2$), respectively:

$$f_1={\displaystyle \frac{\sqrt{\sum _{i=1}^{n_{\mathrm{S}}}{\left({\sigma }_{iso}-{\sigma }_{vm}\right)}^2}}{n_{\mathrm{S}}}}{\displaystyle \frac{1}{{\sigma }_{iso}}}$$

(16)

$$f_2={\displaystyle \frac{\sqrt{\sum _{i=1}^{n_V}{\left({\sigma }_{iso}-{\sigma }_{vm}\right)}^2}}{n_{\mathrm{V}}}}{\displaystyle \frac{1}{{\sigma }_{iso}}}$$

(17)

where ${\sigma }_{iso}$ is the uniform-resistance stress used as a target, ${\sigma }_{vm}$ is the von Mises stress evaluated on each node, and $n_{\mathrm{S}}$ and $n_{\mathrm{V}}$ are the number of nodes on the surface and volume, respectively. Therefore, $f_1$ and $f_2$ are the root mean square deviations of the von Mises stress evaluated in all surface nodes and in all volume nodes, respectively.

Additionally, an energy factor $f_3$ was introduced, defined as the ratio between the elastic strain energy and the maximum energy that could be accumulated if all points are stressed at the maximum stress value [9]:

$$f_3={\displaystyle \frac{\xi \ast 2E}{{{\sigma }_{iso}}^{2}V}}$$

(18)

where $\xi$ is the accrued elastic strain energy, $E$ is the Young’s module and $V$ is the structure’s volume.

3. Test Cases

In order to test the workflow, two cases of increasing complexity were considered, which are reported in the following paragraphs. Case 1 is a generic frame structure; Case 2 is a lattice-like structure.

3.1. Case 1

3.1.1. Baseline

Case 1 is a generic frame structure. The initial geometry and dimensions are shown in Figure 4. Nodes 1, 5 and 9 are fixed, and a vertical force of 7 kN is applied at node 12.

A 3D CAD model of this baseline configuration was created from the diagram. It should be emphasized that the model, being extremely simple, consisting of a series of beams with a constant circular cross-section, can be easily generated with a scriptable CAD editor. Specifically, Ansys Spaceclaim^® (v. 2021 R2) was used to automatically generate the baseline CAD.

Finally, the FEM model and von Mises stresses are shown in Figure 5 and Figure 6.

3.1.2. Step 1: Analytic Optimization

Analyzing the von Mises stresses, it can be seen that the material is not exploited to the fullest extent; in fact, there are unloaded zones and stress peaks at the joints. The analytical method described was applied to optimize the beam section by imposing a uniform-resistance stress of 250 MPa, resulting in the geometry shown in Figure 7.

It can be seen that the geometry obtained has two limitations for a CAD representation:

• There is no continuity of section in the joining zones of two or more elements;
• There are areas of zero cross-section.

Mesh morphing was used to create the FEM model of the analytically optimized geometry. From Timoshenko’s beam theory, by imposing the uniform strength condition, the analytical relation of the optimal section can be obtained (Equation (10)). Equation (10) is computed in the local reference system of each beam element, and originating at the node with the lowest node index.

Noting this correlation, it is possible to create a point cloud from the initial geometry (constant radius section) to the optimized geometry. At this stage, the only critical aspect to consider is the transition from the local reference system of each beam, in which the correlation of R(x) to the global geometry is defined. Furthermore, in order to have good mesh quality and to avoid the problem of continuity in the connection zone, buffers are created between each beam, in which no points are defined. Finally, a minimum section is set. Figure 8 shows the point cloud used for this case, Figure 9 shows the deformed shape.

This point cloud is used as an RBF field. In fact, using RBFs, the displacements defined on the source points are mapped onto the mesh nodes. In this way, the optimized geometry can be obtained automatically.

Analyzing the von Mises stresses (Figure 10), it can be seen that there is a better utilization of the material in the areas away from the neutral section, where the stress is about equal to the imposed uniform-resistance stress; however, there are problems in the striction zones and near the fittings, for which the analytical method used does not give a valid solution.

3.1.3. Step 2: BGM

In order to solve problems at the edges where the Timoshenko beam theory is not valid, and thus to achieve an optimized 3D geometry, the BGM was used. The BGM was only applied in the critical zones, i.e., zones of striction and junction between several elements. The following image compares the geometry of step 1 and the final geometry obtained with the BGM. It is emphasized that the BGM is an iterative method, so in each step, the mesh is updated according to the measured surface tension. It is evident, therefore, how starting from an already-optimized geometry and intervening only in certain areas allows the time required for optimization to be considerably reduced. Figure 11 and Figure 12 show a comparison of step1 and step2 optimized geometries.

Analyzing the von Mises stresses (Figure 13), it can be seen that, overall, there are no major differences, but the main criticalities are resolved: material is added in the squeeze zones and the fillets are smoothed, and excess material is removed (especially in the z-direction), resulting in organic and more efficient shapes.

3.1.4. Comparison

To compare the results, the values of f1 at each node and the scalar values of the factors f₁, f₂ and f₃ were compared (Equations (16)–(18)) (Figure 14). From the contours of f₁, it can be observed that from baseline to step 1, the material utilization improves greatly, but there are some problem areas that are completely solved in step 2. Analyzing the scalar values in

Table 1. Comparison of stress factors.

	VMmax [Pa]	$\mathbf{Vol} \left[{\mathbf{m}}^3\right]$	f1	f2
Baseline	6.9 × 108	4.16 × 10−3	0.72	0.8
Step 1	1 × 109	2.24 × 10−3	0.53	0.64
Step 2	5.7 × 108	1.95 × 10−3	0.48	0.612

and

Table 2. Comparison of energy factors.

	Strain Energy	f3
Baseline	48,221 J	0.075
Step 1	98,108 J	0.28
Step 2	10,223 J	0.335

, a drastic reduction in the f₁ and f₂ factors can be observed, and the volume (and thus the weight) decreases a lot. The energy factor also improves greatly, due to the combined effect of the volume reduction and the increase in deformation energy.

3.2. Case 2

3.2.1. Baseline

The second case analyzed is a lattice structure. Only one layer was considered and a pure shear condition was applied Figure 15.

Analyzing the von Mises stresses (Figure 16), it can be seen that some areas are unloaded, while some stress peaks are present. Therefore, the material is not fully utilized, and we are very far from a condition of uniform resistance.

3.2.2. Analytic Optimization

As in the previous case, the analytical method was applied (Figure 17a), obtaining a correlation between the radius of each beam and the x-axis along the beam axis (Equation (19)). This correlation was used to create the point cloud (Figure 17b) to obtain the optimized geometry. As in the previous case, buffers were created between two successive beam elements, and a minimum section was set. RBF points were used to deform the mesh (Figure 17c).

Looking at the von Mises stresses (Figure 18), it can be seen that excluding the neutral section zone, the stresses are close to the imposed iso-resistance value. However, there are critical zones with stress peaks at strictions and junctions of several elements.

3.2.3. Step2: BGM

As in the first case, the BGM was used to resolve critical areas and further optimize the geometry, adding material where the surface tensions were very high and conversely removing material where it was not needed. A comparison of step 1 and step 2 geometry is shown in the Figure 19. It can be seen how the BGM acts mainly in the strictions and fillets, resulting in a more organic geometry, with less volume (and therefore weight), and much-reduced tension peaks (Figure 20).

3.2.4. Comparison

To compare the results, the nodal values of f₁ and the scalar values of factors f₁, f₂ and f₃ were analyzed. The f1 contour plots (Figure 21) show a significant improvement in material utilization from the baseline to step 1, with some problem areas fully resolved in step 2. A closer look at the scalar values in

Table 3. Comparison of stress factors.

	VMmax [Pa]	$\mathbf{Vol} \left[{\mathbf{m}}^3\right]$	f1	f2
Baseline	2.2 × 108	7.9553 × 10−6	0.85	0.87
Step 1	7.8 × 108	2.8534 × 10−6	0.6	0.69
Step 2	4.2 × 108	2.3051 × 10−6	0.55	0.64

and

Table 4. Comparison of energy factors.

	Strain Energy	f3
Baseline	1.8075 × 10−2 J	0.014
Step 1	9.1444 × 10−2 J	0.2
Step 2	0.10943 J	0.3

reveals a substantial reduction in both $f_1$ and $f_2$, accompanied by a significant decrease in weight. The energy factor also improves considerably, thanks to the combined effect of volume reduction and increased deformation energy.

4. Conclusions

This paper introduces a novel two-step workflow that integrates both 2D analytical and 3D numerical methods for the optimized design of frame structures. By leveraging the strengths of each approach, the proposed method efficiently addresses the challenge of achieving uniform strength while minimizing material usage and weight. The initial step, based on the Timoshenko beam theory, rapidly generates an optimized geometry for the entire structure using an analytical model. This method is highly effective for uniform-strength profiles, but is limited by edge effects in regions where three-dimensional phenomena, such as strictions and joint zones, play a critical role.

To overcome this limitation, the second step incorporates the Biomechanical Growth Method (BGM), a bio-inspired numerical approach that refines the optimization process in regions affected by triaxial stress fields. This method ensures that areas near boundaries and joints, where 2D analysis may fall short, are optimized for strength and stability, while maintaining the structural topology. The combined approach thus ensures a comprehensive optimization across the entire structure, balancing both efficiency and accuracy in areas of complex stress distribution.

The results of the test cases are promising, demonstrating improved efficiency in terms of stress distribution and energy factors. Optimized geometries generated using this hybrid method exhibit superior performance compared to those designed solely through traditional approaches. This dual approach holds significant potential for advancing the design of frame structures, offering a reliable and efficient tool for engineers in the field of structural optimization, and enabling the development of an automated tool that can input any frame structure and output the optimized geometry ready for additive manufacturing.

The analytical model currently applies to planar frame structures, but can be extended to 3D structures by following the same idea.