Optimization on compressive strength of sandwich plates with internal arborescent microstructure

Abstract

The subject of this work is the optimization of a topology for sandwich plates with an internal arborescent microstructure. Here, we propose a parametric approach to the optimization using visual programming methods in the Dynamo Sandbox environment. An internal structure of the sandwich plate was optimized in terms of minimizing the weight of the entire structure in relation to the compressive strength. Several variable parameters in the optimization process were applied: lengths of individual tree levels, their cross-sections and locations of nodes. To validate obtained results of the optimization process, a 3D model of the plate was printed and tested on compressive strength. The results were also analysed and compared with other authors’ research.

Keywords

Sandwich plates optimization intelligent materials parametric design bionic

Introduction

Taking advantage of nature-based solutions in a field of technology is not a novel concept. Humanity has been looking for inspiration to design and create new structures based on the observation of nature since the times of Leonardo da Vinci.¹ Biological structures evolve over millions of years, so this approach seems to be just. They are naturally adapted to specific environmental conditions. In many cases, their internal structure and their morphology are highly optimized for the transfer of various forces and stresses. This fact is the basis of the trend known as bionics and has been the subject of numerous studies over the years.^2–4

The works on the study of multiscale structures in this field deserve special attention. They constitute a complicated and multi-faceted answer to the problem of multi-criteria optimization. Wu et al.⁵ presented a review paper on a topology optimization of multi-scale structures. They categorized existing approaches, explained the principles of each category, analysed them, and discussed open research questions. In⁶ authors introduced an approach where design optimization was applied at two scales: the macroscale (structure optimization), and the microscale (material optimization). Jiang et al.⁷ aimed at devising a numerical scheme for designing bionic structures by combining a two-stage parametric level set topology optimization with the conformal mapping method. The obtained structure and metamaterial designs were further synthesized to form a multiscale structure using conformal mapping.

On the other hand, works on “bio-inspired” multiscale structures often present certain mathematical simplifications, and they are considered using methods of fractal analysis. In⁸ the authors present comparative study between Fractal Dimension and Multi-Scale Fractal Dimension in a shape analysis context. Tesar⁹ adopted the bionics and fractal configurations from the nature in structural engineering. He used neural networks corresponding with genetic algorithms. Md Rian and Sassone¹⁰ briefly discussed the biological functions and the mechanical properties of trees with regard to their fractal-like complex shapes and applications in architectural structures.

Depending on the specific needs and subject of the considered issue, scientists within the so-called bionics, draw an inspiration from various areas of nature. One of the most common types of “bio-inspired” structures are tree-structures. Wu et al.¹¹ proposed a new form-finding method (inverse-hang recursive method) for branching structure. They also applied it in the form-finding analysis of a practical branching structure and proved its practicability. Others focused specifically on branching columns, which proves their effectiveness as structures subjected to compression. Dekhn and Shadhan¹² investigated the failure load, vertical and lateral displacement, and failure mode of tree-like columns subjected to a combination of axial and lateral loading using FEM. Khamees and Shadhan¹³ conducted an experimental study to find the effect of the branch’s height to total height and specimen’s width to total width ratio on the structural behaviour of a tree-like steel column. There are also a number of studies on optimization techniques of tree-structures. In¹⁴ the authors presented a novel shape and topology optimization method to find optimal branching structure to support a roof with a specified geometry. Zhao et al.¹⁵ determined the optimal topology of branching structure with an algorithm that removes branches according to the efficiency of components, and finally achieves the optimization goal of each level.

The subject of this study are plates – structural elements working mainly on bending and compression. This led us to some specific inspirations and narrows the huge potential area of structure topology for arborescent elements. The above-presented literature review and the latest research using generative design shows that many optimized topologies are tree-structure.¹⁶ It combines relatively high flexibility, lightness and strength, transferring compressive stresses in a very effective way. On the other hand, optimal structure in terms of bending often takes the shape of an I-section beam or sandwich panel (Figure 1). The amount of material in the middle part of this structure is deliberately reduced to obtain an element that is lighter, but strong at the same time. This is the result of the extensive layers in the upper and lower parts of the structure – they transmit compressive stresses or stretching.

Another interesting proposal modelled on bionic solutions is the honeycomb structure, which also combines strength with lightness. Cheng et al.¹⁷ increased the strength of a honeycomb structure under a static load using an innovative and integrated multi-objective optimization procedure. In¹⁸ honeycomb sandwich panels were tested on the local buckling tendency and experimentally evaluated with respect to various design variables. An optimization problem in honeycomb structure was also presented in¹⁹, where the optimization process is performed using a set of algorithms including the gravitational search algorithm (GSA).

Therefore, this paper proposes a solution that combines various aspects of bionics and is also based on the achievements of the aforementioned works. It combines the simplicity of the approach, the engineering capabilities of the real design and implementation of a specific sandwich plate microstructure. The proposed structure is a sandwich panel, where the internal structure is symmetrical arborescent structure and “trunks” are in plane of an internal layer. This layer is subject of optimization procedure. External layers are not optimized, and they are only “tree crown” support. The inspiration of bionic led the authors to honeycomb structure, which, on the one hand, is an applicable connection of the tree branches ends (the internal structure), and on the other hand, it is lighter than the standard solution of the full outer layer. The “branches forks” are in a multi-level direction towards the plane of the upper and lower plates (Figure 2).

The entire sandwich structure was represented and analysed using truss finite elements subjected only to compression-tension, in accordance with common structural assumptions. Consequently, it is possible to make a significant numerical simplification. Presented approach is less complex than the topological optimization proposed in many studies. Some of these works combines the finite element method in a three-dimensional form with artificial intelligence algorithms²⁰ or evolutionary algorithms.^21–23 An additional advantage is the fact that this type of plate is easier to make, whether in 3D printing or prefabricated technology. As a side note, the post-optimization stage, which allows for construction of the real structure in case of optimization of a topology based on volumetric finite elements, is a complex issue and requires many simplifications and idealization.²⁴

Subject, purpose and scope of work

The subject of this study are light plates with a-priori assumed internal arborescent microstructure (Figure 2(a)). The designed plate consists of two surfaces based on honeycombs geometry (Figure 2(b)). Between the plates there is a symmetrical, multi-scale and multi-level arborescent structure with elective number of levels n (Figure 3(a)). Based on inspiration from biology, it was a priori assumed, that at each subsequent level, except for the last n-th level of “tree crown”, it branches out to 4 “boughs” and at level n it branches out to 6 “branches”. Such a structure, which is a bionic interpretation of trees in nature, regardless of the adopted number of levels n, will always allow to connect “branches” with the honeycomb outer layer structure. It also gives an opportunity for further optimization of the height of individual levels and other parameters, such as cross-sections of “branches” at each level.

The aim of this work is to optimize the above-described structure in terms of minimizing the weight of the entire structure in relation to its compressive strength. Objective function was defined as:

F = \frac{G}{P},

(1)

where G is the mass of tree-structure (the internal, optimized layer), and P is the total uniform compressive load that causes its failure.

Optimized design variables could be heights of each tree levels $h_{1}, h_{2}, \dots, h_{n}$ , cross-section areas of each level of “branches” $A_{1}, A_{2}, \dots, A_{n}$ and normalized eccentricities of the horizontal spacing of the forks at each levels ${\bar{τ}}_{2}, {\bar{τ}}_{3}, \dots, {\bar{τ}}_{n - 1}$ (Figure 4). An eccentricity was introduced for simplification, which is identical and simultaneous in both horizontal directions, x and y. Geometric limitations of these parameters are: $\sum_{i = 1}^{n} h_{i} = H$ , where H is a predetermined height of the entire structure and ${\bar{τ}}_{i} \in 〈 - 1,1 〉$ . Strength limitations are: $⋀_{i = 1}^{n} σ_{i} = \frac{N_{i}}{A_{i}} < k$ , where N_i is normal force in the “branch” of i-th level and k is compressive strength of the material.

An additional goal of the work is to analyse the process of practical implementation of an optimized structure by its 3D printing and comparing the obtained results of strength tests with theoretical predictions derived from modelling in finite element method software. The work covers the stage of structure optimization, the stage of theoretical solution idealization, its transformation into a real physical model, and the stage of strength experimental tests.

Optimization methods

Individual structures and their elements were represented in Dynamo Sandbox²⁵ (Figure 5). It is a visual programming language environment (VPL), which allows for parametric modelling of the geometry of any bar or shell structure. Then, using the Python language, an original instruction block was created using the finite element method. It was possible to calculate the value of the objective function, which was the amount of material for the entire bar structure.

To create the matrix of the finite element method, truss elements with 3 degrees of freedom per node and linear shape functions were used. The global OXYZ coordinate system was introduced in Figure 6. Then the nodal displacement vector was defined in the global coordinate system (2):

u^{e} = {[\begin{array}{l} \begin{array}{l} u_{i x} & u_{i y} & u_{i z} \end{array} & \begin{array}{l} u_{j x} & u_{j y} & u_{j z} \end{array} \end{array}]}^{T} .

(2)

The system of FEM equations for a single finite element was defined as:

{k^{e} u}^{e} = q,

(3)

where q is the load vector. The classic finite truss element was used for calculations:

k^{e} = \frac{E A}{L} [\begin{array}{l} H & - H \\ - H & H \end{array}], H = [\begin{array}{l} C_{X}^{2} & C_{X} C_{y} & C_{X} C_{Z} \\ C_{X} C_{y} & C_{Y}^{2} & C_{Y} C_{Z} \\ C_{X} C_{Z} & C_{Y} C_{Z} & C_{Z}^{2} \end{array}], m^{e} = A ρ L,

(4)

where:

k^{e}

– element stiffness matrix,

H

– cosine matrix of the element,

C_{X} = \frac{L_{X}}{L}

C_{Y} = \frac{L_{Y}}{L}

C_{Z} = \frac{L_{Z}}{L}

L = \sqrt{L_{X}^{2} + L_{Y}^{2} + L_{Z}^{2}}

m^{e}

– element mass used to calculate the final objective function,

E

– Young's modulus, A – element cross-section area, L – element length,

ρ

– element density,

L_{X}

L_{Y}

L_{Z}

– element projection length on the coordinate system axes.

Then matrix (3) was assembled running through all m finite elements (5):

K = \sum_{a = 1}^{m} {T_{a} k}_{a}^{e} {(T_{a})}^{T},

(5)

where

T_{a}

are transformation matrices of individual elements defined as matrices rectangular matrices of size 6 x 3 m. The elements matrices contain a total of six values “1” located in the positions corresponding to the numbers of individual nodes in global notation and “0” in all other cells.

To solve the resulting system of equations of the form (6):

K U = Q,

(6)

where Q is global vector of loads for entire structure, and U is global vector of displacements.

The mathematical package Math.NET²⁶ available in the DynamoBIM environment and Python was used for calculations. By solving the problem (6) using FEM, it was possible to find displacements of individual nodes and consequently – changes in length (strains) of individual tree “branches” $ε_{a}$ . It was applied for each considered geometry and for a predetermined constant external load.

Based on them, it was possible to find stresses in individual “branches”, using the classic formula (7):

σ_{a} = E ε_{a} .

(7)

Then, the built-in generative design tool in the Dynamo Sandbox environment was used. It allows the user to find the optimal solution with a combination of several methods. The first one is searching through the entire space of design variables. This method allows to find the global solution, but unfortunately it is very slow, because all cases from the entire multidimensional domain of design variables are tested. The second method implemented in this environment is fully random search. This method takes number of randomly selected geometries as the only parameter and selects the vector of design parameters, but it does not guarantee proper results. The third method is the use of genetic algorithms built into the Dynamo Sandbox environment. The disadvantage of this method is the lack of detailed information from the developers of this tool, regarding the exact genome coding solutions, crossover parameters, mutations, etc. The fourth and the last method is searching for solutions near specific values of design variables. This method is effective for increasing the accuracy of the solution obtained by other methods. It is also possible to set any restrictions on the input and output parameters of the model, in particular on the allowable structure stresses and deformations.

To find global objective function (1) minimum under defined geometrical (firstly, the sum of the heights of all levels is fixed and constant, secondly dimensionless eccentricities can take fractional values) and strength conditions (the compressive stress in each “branch” of the tree must be lower than the compressive strength of the material), a combination of the first method (searching the entire space of solutions with a relatively rare division) and the fourth method (searching for a solution near a specific point in the space of input parameters) was used. The methods were chosen after performing a number of numerical experiments.

Applied method of solution idealization and 3D printing

The current optimization process has been of a general and universal nature. Therefore, it is possible to apply any method of creating a real structure based on the optimized values of design parameters. In case of this work, 3D printing technology was chosen due to its relatively low cost and ease of model implementation. The 3D printer available for the authors had a maximum print area of 20 cm × 20 cm × 18 cm. Therefore, optimized structure was scaled to these dimensions. Linear and proportional scaling of all geometrical quantities were applied in the optimized model. Next, the model was refined in terms of node connections and the size of sections converging at individual levels of the structure. The simple and direct usage of bars with a constant cross-section caused the errors shown in Figure 7(a), which would cause technological problems during printing. The model was printed layered starting from the “branches” towards the “trunk”. In case of attempt of printing the model presented in Figure 7(a), there would be problems with supporting the small pieces of nodes marked in red. Therefore, chamfering the heads of individual “branches” was necessary, so that the cross-sectional area of the bar reaching the node (fork) from the bottom was adjusted to the cross-sectional area of the connected higher-level “branches” (Figure 7(b)).

Before the final application of this solution, a precise FEM model with volume elements was created. Using a test problem (uniform and axial loading of a thicker “branch”, hinged support of thinner “branches”), the impact of the proposed phasing of rods on stress values in the joint was analysed. A detailed analysis, presented in Figure 8, proved that the use of phasing for the thicker rod near the node does not increase stress values, but only reveals more strained areas by eliminating material from regions that do not participate in stress transmission. After analysing the obtained numerical results, the method of chamfering bars near nodes for each printed model was chosen.

Material tests

In addition, in order to investigate the suitability of this material for model manufacturing, test samples were printed – cubes with dimensions of 3.0 cm × 3.0 cm × 3.0 cm, which were subjected to a compression test. Two series of tests were conducted independently for Z-GLASS and Z-HIPS materials, from which the models were created. The printing direction was considered, destroying two samples made of this material, one was compressed in the direction parallel to the printing plane, and the other in the direction tangential to this plane (Figure 9).

As a result of the conducted research, it was found that the Z-GLASS material exhibits practically complete isotropy of mechanical properties, and the compressive strength of the material on cubic samples with dimensions of 3.0 cm x 3.0 cm x 3.0 cm was determined to be 50 MPa (Figure 11). None of the samples delaminated (Figure 9(c) and (d)). As shown in Figure 10(d), the sample compressed in the direction parallel to the print plane delaminated, resulting in significantly lower strength (Figure 11 – orange line) than in the case of compression in the direction perpendicular to the print direction (Figures 10(c) and 11 – blue line). Finally, the compressive strength values for Z-HIPS material were obtained, equal to 53.8 MPa (perpendicular to layering) and only 31.5 MPa (parallel to layering).

Research examples

The three research examples were analysed: I. the 3-level structure of a single tree, II. the 4-level structure of a single tree, III. symmetrical 3-level plate structure. All three examples were subject of the optimization procedure described in the previous chapters. However, in the case of examples I and III, the procedure of solution idealization, 3D printing and practical verification of the actual physical strength of the structure in a testing machine were also used.

The 3-level structure of a single tree

The first of the numerical examples is optimization of a single tree-structure. This is the first step in creating a plate with an arborescent structure in example III. Three levels of tree branches were assumed, which leads to five independent optimization parameters – two related to the levels of individual branches: h₃, h₄ and three related to the cross-sections of individual branches: A₂, A₃, A₄. Constant values were assumed for the parameters of the total height H =10 cm and the size of the honeycomb side a = 1 cm. Using the generative design tools described above, the following set of optimization parameters was obtained for which the value of the objective function (minimizing the mass of the tree bar structure) was the smallest:

A_{2} = 12,28 [-]; A_{3} = 4,07 [-]; A_{4} = 2,08 [-]; h_{3} = 1,57 cm; h_{4} = 0,63 cm .

It is worth noting that in the case of parameters related to the size of the cross-sections of individual tree branches, dimensionless and normalized values were optimized. This solution was convenient for numerical reasons and universal because it was not necessary to consider the accuracy and precision of the 3D printer model during optimization, which would have been quite challenging in practice. It should also be noted that this solution makes it possible to obtain an optimized structure with the required load capacity expressed in kN/m² by applying cross-sectional scaling while maintaining proportions between individual cross-sections.

In the next step, post-processing related to the realization physical model using a 3D printer was performed. The model was scaled in terms of the available 3D printer and the selected material, and the diameters of the bars of individual “branches” were adopted as: ∅2 = 1 cm, ∅3 = 0.4 cm, ∅4 = 0.2 cm. The values of individual diameters expressed in cm were selected to, on the one hand, preserve the previously calculated optimal proportions between individual “branches” as accurately as possible, and on the other hand, to ensure a realistic possibility of printing the model tailored to both the device used and the printing material. Thus, the diameter of the thickest branch was a priori set to 1 cm, and then the smaller branches were proportionally scaled accordingly, rounding the results to 1 mm for technical reasons:

\emptyset_{3} = \frac{A_{3}}{A_{2}} ∙ 1 c m = 0,33 c m \approx 0.4 cm; \emptyset_{4} = \frac{A_{4}}{A_{2}} ∙ 1 c m = 0,17 c m \approx 0.2 cm .

(8)

Z-GLASS material was used²⁷ with Young's modulus of 4.12 GPa and theoretical tensile strength of 76.5 MPa.

Then, the FEM model of the designed structure was created in the Autodesk Robot Structural Analysis software, obtaining a tensile stress in the honeycomb layer equal to 11.22 MPa (Figure 11(a) and (b)) and 33.08 MPa for compressive stress. The boundary conditions and the applied load of 1 kN to the entire model were shown in Figure 12(b). Considering the elastic strength of the Z-GLASS material, defined during the experiment – 50 MPa, this leads to a theoretical breaking force of – 1654 N.

Finally, the strength test was performed using the Zwick Roell type BT1-FR050TH.A1K machine (Figure 13(a)). Obtained value of the breaking force was 2110 N, which is 127% of the theoretical strength calculated based on preliminary compression strength tests of the material. The actual strength of the model may be higher than the theoretically calculated value due to the real elastic-plastic characteristics of the Z-GLASS material (Figure 14). In the model, it is probable that plastic deformation occurred in the most stressed elements and, as a consequence, stress redistribution occurred, which increased the final result of the destructive force value.

To compare the real model’s failure scheme with a numerical model, another FEM model was created, this time based on volume finite elements. Uniform loading was applied from the top and hinged support of the tree trunk circumference. Stress maps reduced according to the Mises hypothesis are presented in Figure 13(c). The model's failure form (Figure 13(b)) corresponds with the theoretical maximum stress distribution in the analytical model (Figure 13(c) and Figure 15(b)). The structure ultimately collapsed by breaking at the nodes of the most heavily loaded, outer branches of the second level.

The 4-level structure of a single tree

In the second example, the possibility of optimizing a more complex 4-level tree-structure was analysed. Theoretically, it leads to 7 independent parameters, i.e., 3 related to the levels of individual branches: h₂, h₃, h₄ and 4 related to the cross-sections of individual branches: A₁, A₂, A₃, A₄. However, after the analysis of the initially obtained optimization results for this structure two additional parameters were used. Additional non-zero horizontal eccentricities ${\bar{τ}}_{1}$ and ${\bar{τ}}_{2}$ of ‘branching” at levels 1 and 2 were introduced, defined according to Figure 4. As a result, a better optimized structure with a lower value of the objective function was obtained.

Finally, after the optimization process, the following parameter values were obtained:

A_{1} = 5.34 [-]; A_{2} = 1.20 [-]; A_{3} = 0.61 [-]; A_{4} = 0.05 [-]; h_{1} = 1.87 cm; h_{3} = 1.83 cm; h_{4} = 1.09 cm {\bar{t}}_{1} = 0.37 [-]; {\bar{t}}_{2} = 0.08 [-]

It is worth noting that the obtained 4-level structure had 2.34 times less mass (the value of the objective function) compared to the 3-level structure. Then, the possibility of creating a physical model based on the theoretical optimized scheme was analysed, and due to the large number of small elements, further attempts to 3D print this model were abandoned.

Four units of tree structures

The last analysed example was the structure of a symmetrical plate based on an optimized 3-level tree. To create it, the previously optimized 3-level structure described in Example I was used. A model of asymmetrical plate was created based on 2 x 2 x 2 = 8 individual modules of the tree described in Example I (Figure 16). Then, it was scaled to the actual size: 17.2 cm x 19.5 cm x 15.1 cm. Due to the very small elements of the last level, the Z-HIPS material was used²⁸ to make the 3D print. The total weight of the model was 210 g.

Half of the model was used, because employing the full model would have required precise alignment and bonding of both parts (top and bottom of the plate had to be printed separately), which could introduce an additional factor of uncertainty in the results for the strength test in the Zwick Roell machine type BT1-FR050TH.A1K. In order to evenly distribute the load, a steel plate weighing 3.12 kg was used. As a result of the experiment, the destructive force value was 4131 N (Figure 17(b)). The elements of the second level were destroyed (Figure 18).

The obtained destruction form (Figure 18) is consistent with the predictions of the FEM model – the most stressed are the branches of the second level near their lower nodes. The specific obtained value of the destructive force is difficult to verify due to the anisotropy and clearly nonlinear behaviour of the Z-HIPS material (Figure 11– blue and orange lines). The estimation obtained by extrapolating the previous model, taking into account the scaling factor between the models equal to 0.72, using 4 trees in the layered structure of the plate and considering the fact that the model was loaded perpendicular to the printing plane (higher compressive strength of 53.8 MPa), caused in a result of 3690 N. Therefore, the actual destructive force was again higher than the theoretically determined by 12%. This may result, as in the previous case, from the elastic-plastic characteristics of the actual material, while the FEM numerical model considered only the elastic characteristics. Finally, the ratio of the destructive force (F_c, [N]) to the weight (F_m, [N]) of the final physical model was calculated as 1967:1 (for Z-HIPS) and 2344:1 (for Z-GLASS). The obtained values were attempted to be compared with data obtained by other authors. However, this is very difficult due to the specificity of each study, the method of loading, the diverse nature of models, etc. No research on the strength of similar plates with an internal tree-like microstructure was found in the literature. The most similar studies are presented in²⁹, where the authors present three types of 3-D printed lightweight polymer-based sandwich cores. These sandwich structures are made of polypropylene-flax bonded to the polylactic acid lattice structure core using bi-component epoxy adhesive. In the comparison, two of the three core types from,²⁹ that most closely resemble the structures studied in this study were used (lattice 1 and lattice 3). The authors presented their results of compressive out of plane tests as normalised maximum load (compressive force [N] referred to mass [g]), so the values were converted from mass to weight. Obtained results are listed in Table 1.

Table 1.

Comparison on compressive destructive forces for arborescent sandwich plate developed in presented study and structures found in the literature.²⁹

Reference work	Structure	F_c [N]	Mass [g]	F_m [N]	F_c:F_m
²⁹	Diamond/tree-like design with a vertical strut	—	—	—	509000:1
²⁹	Diamond/tree-like design without a vertical strut	—	—	—	230600:1
Present work	Arborescent sandwich plate (Z-HIPS)	4131	210	2.10	1967:1
Present work	Arborescent sandwich plate (Z-GLASS)	2310	90	0.90	2344:1

Although the obtained value of the F_c:F_m coefficient for the considered structure is two orders of magnitude lower than the values obtained in 29, it should be noted that the model considered in this study is a much larger structure and was printed on a standard 3D printer using widely available and inexpensive materials. The differences in loading conditions and the nature of the model's operation also work to the disadvantage of the plate structure with an internal tree-like structure, which means that the results presented in Table 1 are only of indicative value.

Summary and conclusions

In this work, a parametric design of the arborescent sandwich plate structure was carried out. It was optimized using generative design tools and was also practically implemented by 3D printing technology. The model was printed with proper accuracy and attention to detail, and the defects and surface errors created during the printing process were practically imperceptible and most likely did not have any impact on the course of the strength tests.

As a result of the conducted research, the following conclusions were made:

1. Used programming tools, i.e., the Dynamo Sandbox environment, enables effective design and optimization of theoretical bar-plate structures.

2. Use of generative design tools in the Dynamo Sandbox environment combined with special programming solutions in Python, allowed for effective optimization of the structure with up to 9 independent parameters.

3. Theoretically designed structures can be effectively implemented using 3D printing technology.

4. The method of destruction of the printed structures was consistent with the predictions of the FEM model based on volumetric elements.

5. In destructive tests of physical models, the obtained values of the destructive force 10-20% higher than the theoretically calculated force using the FEM model, which is likely due to the elastic-plastic behaviour of the actual material of the model.

6. During the compression test, Z-GLASS material exhibited complete isotropy, while Z-HIPS material delaminated under compression in the direction parallel to the print plane, demonstrating significant anisotropic properties.

7. The final obtained ratio of compressive force to the mass of the plate structure was 2000:1. The best way to compare obtained results reliably, would be to create a solid plate printed with exactly the same materials. In the future, we plan to perform this type of research to expand this optimization issue.

The obtained arborescent structure might be used in the future as a skeleton of plates constituting the insulation of ceilings and floors, in places where the traditional solutions used so far were characterized by insufficient compressive strength. In the future, there are plans to extend the research to plates with varying thickness, where the boundary layers are not flat surfaces. It is also possible to consider the use of programming tools, developed and tested for the purposes of this paper, for other optimization applications of any bar or plate structure.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Artur Wirowski

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