Topological optimization analysis aimed at minimizing the mass of a hook fixed some design parameters, such as geometric constraints and load conditions.

The following article is an extract from a project carried out for the course of Advanced Techniques for the design of Prosthetic Devices. Held for Medical Engineering at the University of Rome Tor Vergata .

Authors: Mastrofini Alessandro & Muscedere Erica

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This topological optimization project involves the optimization techniques applied in Solidworks, based on the SIMP algorithm, and several theoretical points necessary to design an object that can be printed correctly with FDM techniques so that it also maintains good mechanical properties.

Topological optimization

Topological optimization involves analyzing the entire domain which is being questioned. The goal is to remove or leave areas of material until the optimal condition is found and such as to satisfy constraints and requirements.

The method typically implemented in commercial applications is the structural penalty method or SIMP. This method involves assigning a fictitious Young’s modulus to each element of the domain (discretization) by means of a virtual density:

E\left(\rho_{e}\right)=\rho_{e}^{p} E_{0}

So each element will have a new stiffness:

K_{S I M P}=\left[\rho_{\min }+\left(1-\rho_{\min }\right) \rho_{e}^{p}\right] K_{e}

We start by defining a computation domain for which we are sure of the fulfillment of the constraints and requirements (eg minimum thicknesses, symmetry relations, maximum stresses, maximum displacements). The target is then chosen:

  • Maximization of the stiffness-to-weight ratio
  • Lower maximum displacement
  • Lower mass
  • Maximization of stiffness

Finally, we obtain a result where each element corresponds to a certain fictitious density. Through a threshold, the elements considered ‘ineffective’ are cut away for the purpose of optimization and the rest are filtered to avoid the effect of a stepped surface (due to the cut mesh).

Additive manufacturing

3D printing is an emerging technology (sometimes even abused) that allows the creation of parts and assemblies using additive logic (by adding material a little at a time). It differs from the main conventional technologies which are generally based on a subtractive logic (chip removal) or molding.

Compared to conventional molding technologies, it allows to avoid mold and mold holder costs, making it particularly suitable for the construction of prototypes or limited series (pre-series products where a few hundred objects are produced). In general, additive technology allows the creation of even very complex shapes, overcoming the limits of conventional technologies.

FDM technology, filament deposition printing, is certainly among the most popular for its simplicity and low costs of equipment and materials. Accessible in almost all prototyping labs and also for makers. It is particularly suitable for the use of widely used thermoplastic materials such as ABS and PLA. More advanced technical materials are also used. It allows the creation of aesthetic and functional components.

Preliminary analysis

To tackle a topological optimization analysis, the first step is to make sure that the project requirements are met, at least in the starting domain. In particular, the structure in question must withstand a load of 12.85 N and preserve some areas for weight attachment and assembly. The piece will be produced via 3D printing in Z-ABS. This material, owner of Zortrax, has slightly different properties from the classic ABS filaments (table 1).

Once the component was built, a first static analysis was carried out to characterize the intensity of the stresses following the design load (fig. 1). Interlocking constraints on the top of the back face are considered. That is, the back face has been split approximately 1/3 with a weed line (this will be specified better in the topological optimization section).

Fig. 1: Analisi preliminare: carichi e vincoli (a); tensioni nel caso di solo incastro (b); tensioni nel caso di incastro + carelli (c)
Fig. 1: Preliminary analysis: loads and restraints (a); stress in the case of joint only (b); stress in case of interlocking + carts (c)

From the results it can be seen that the maximum stress is more than an order of magnitude below the yield point. Furthermore, the type of constraints considered show how the lower part would be free to slide backwards, greatly increasing the stress on the separation edge between the upper and lower part (fig. 1b).

Stiffness and structural constraints

In this regard, in the subsequent analyzes, trolleys on the rear face are considered, representing the rear bracket present in the project requirements. However, the carriages allow vertical sliding so as not to overestimate the overall stiffness (fig. 1c). Therefore, also considering the impossibility of the rear face to go backwards, the tensions in the structure are even lower. There is certainly a way to proceed with topological optimization.

Elastic module1.08 GPa
Poisson’s coefficient0.394
Density1195 kg/m3
Yelding stress25.89 MPa
Tensile stress30.46 MPa
Tab. 1: Properties for Z-ABS

Topological optimization analysis

A first optimization was carried out in a more coarse way to understand which areas of the structure were necessary to support the load.
To respect the design constraints it is necessary to add regions to be preserved on the cylindrical holes and on the lateral regions fig. 2a. A plane of symmetry can also be added.

Considering a joint at the level of the screws, the stiffness of the structure is overestimated and the optimization produces two separate bodies (fig. 2b). More realistic constraints can be hinge-type constraints on cylindrical holes together with some material connecting the two holes (or the two lateral ends) in such a way as to increase the torsional stiffness (of which the two separate bodies would be completely devoid). A first level of optimization is present in fig. 2c.

Fig. 2: Prime analisi di ottimizzazione topologica. Zone da preservare (a); risultato approssimativo con due corpi (b); risultato di partenza considerando cerniere cilindriche e piano di simmetria (c)
Fig. 2: First topological optimization analyzes. Areas to be preserved (a); approximate result with two bodies (b); starting result considering cylindrical hinges and plane of symmetry (c)

By analyzing these structures obtained from the first optimization, we can see how we are still far from the limit load and there is no way to optimize.

Optimization constraints

Before carrying out the static analyzes it is necessary to manually recreate the areas to be constrained (fig.3a)

Fig. 3: Analisi statica del primo test di ottimizzazione: ricostruzione delle superfici piane (a); campo di tensione (b). La struttura risulta lontana dal limite di snervamento.
Fig. 3: Static analysis of the first optimization test: reconstruction of flat surfaces (a); stress range (b). The structure is far from the yield point.

It can be seen how the zones set as no-design zones are overestimated for the thickness and could be reduced.
Then we proceed with a second more refined optimization campaign, at least in terms of constraints. A not too fine mesh is still used for the mesh as it is not considered necessary to push the study to the limit for various considerations. First of all, in the design phase, it is necessary to take into account that the object will be 3D printed so the finishing cannot be less than the thickness of the filament (0.4 mm).

Design and production

3D printing affects strength even more importantly as material properties will be degraded and free of isotropy, with much lower strength perpendicular to the layers. Added to this is the fact that by correctly balancing the smoothing and the fictitious density threshold it is possible to obtain a well-defined result.

A vertical load of 13 N (overestimating), symmetry constraint and areas to be preserved was always applied to the new analysis. The limit stress was taken as 20 MPa, about 80% of the yield point but from the static analyzes it can be seen that the tension range is well below this value thanks to the constraints set. The areas to be preserved have been reduced in thickness compared to the previous optimization and a thin area has been added between the two holes in order to guarantee the uniqueness of the body and counteract the torsional effect (which is better discussed in the verification analyzes ).

Furthermore, to reduce the computational cost, an area of ​​material deemed unnecessary to bear the load has been removed (fig. 4a)

Fig. 4: Ottimizzazione finale: struttura di partenza svuotata (a); no-design zone (b); risultato con diagramma a falsi colori della densità fittizia (c)
Fig. 4: Final optimization: starting structure emptied (a); no-design zone (b); result with false color diagram of the fictitious density (c)

The result of this analysis is exported as a solid body and further verification analyzes are carried out.

Checks on topological optimization

The model is then subjected to some verification analyzes.
A first analysis is made by considering the interlocking on the holes and of the carriages on the rear surface. A second analysis, on the other hand, is carried out leaving the structure an extra degree of freedom, i.e. allowing rotation around the holes, imagining the screws are not tightened. Both simulations predict a vertical load of 13N.

Fig. 5: Confronto nei diagrammi a falsi colori delle tensioni nella struttura finale tra la simulazione con incastri (a) e cerniere cilindriche (b). Lasciando la struttura libera di ruotare si vede come il carico induce un effetto torsionale che a sua volta induce una tensione maggiore nella zona superiore. Le tensioni massime risultano inferiori nel caso delle cerniere.
Fig. 5: Comparison in the false color diagrams of the stresses in the final structure between the simulation with joints (a) and cylindrical hinges (b). Leaving the structure free to rotate you can see how the load induces a torsional effect which in turn induces a greater tension in the upper area. The maximum stress are lower in the case of hinges.

In the first two verification analyzes, a maximum stress lower than 50% of the yield point is obtained. The maximum displacements are 0.46 mm in the case of interlocking and 0.51 mm in the case of cylindrical hinges. As can be seen from the deformations in fig. 5 the maximum displacements involve the opposite ends of the load support areas, in the opposite direction.

It can be seen how the simulation in fig. 5a is not fully representative of reality as the cable with which the weight is attached would not allow the ends to widen, rather it induces a further horizontal load which tends to bring the two areas closer together. Further analyzes are then carried out to simulate the projection of this load, even making it extreme.

Safety coefficients

An analysis is carried out with a remote load of 15 N (overestimated) by also loading the side faces and centering the remote point in the plane of symmetry, 3 cm below the model. This tends to simulate the cable with which the weight is attached if it is very short (and therefore the horizontal component of the load is greater). A simulation is also carried out with the addition of a horizontal load of 2N which tends to compress the two load support areas (figs. 6b and 6d), with a view to an accidental load in the test phase, having no control over the experimental verification procedure.

Fig. 6: Analisi statica di conferma. Diagramma a falsi colori delle tensioni con carico remoto (a) e con carichi orizzontali (b); degli spostamenti con carico remoto (c) e con carichi orizzontali (d)
Fig. 6: Confirmation static analysis. False color diagram of stresses with remote load (a) and with horizontal loads (b); of displacements with remote load (c) and with horizontal loads (d)

Therefore, it was also verified by considering the interlocking, or the case in which the screws are well tightened and therefore the torsional effect falls more on the two arms and not on the upper part. In this case (fig. 7a) the structure is even more secure. The analyzes show that the structure is safe, under 50% of the yield stress.

Fig. 7: Analisi statica di conferma con incastro sui fori. Diagramma a falsi colori delle tensioni (a) e degli spostamenti (b)
Fig. 7: Static confirmation analysis with interlocking on the holes. False color diagram of stress (a) and displacements (b)

There is also room for further improvements but to be sure we need more information on how much the material loses in the printing phase and for the adhesion of the layers.


Some areas have been rebuilt to facilitate static analyzes.
To carry out the finishing, the .stl file was loaded in Power Surfacing using the subdivision surfaces to perform a defeaturing and soften the sharp edges (fig. 8). Going to carry out the same static analyzes again, results are obtained that are very similar to the previous ones that we believe satisfy the project requirements.

Fig. 8: Rifinitura. Vista laterale: rifinito (a) e originale (b); vista frontale: rifinito (c) e originale (d); ingrandimento sul gancio: rifinito (e) e originale (f)
Fig. 8: Finishing. Side view: finished (a) and original (b); front view: finished (c) and original (d); magnification on the hook: finished (e) and original (f)

3D printing

For 3D printing, to maximize strength, the placement would be vertical, placing the hook on the side. However, this would lead to high printing times and an excessive number of media. A positioning is then carried out as in figures fig. 9 by rotating the structure about 30 °. This allows you to make the most of the layer while keeping the filaments unique for a greater length.

Fig. 9: Layer position and print stages 25% (a), 50% (b) and 100% (c)

On the other hand, maintaining a rotation of 0 or 90 ° would lead to having the continuity of the filament in the shorter length (lower resistance) in the thin areas. The file for the M200 printer was also generated with the same settings.

3D prototype

The model was then printed (Zortax M200) and tested: it fully meets the project requirements and was the lightest (among the different models participating in the contest).

Modello appena finito di stampare con la presenza dei supporti di stampa
Model just finished printing with the presence of the print media
Struttura risultante dall'ottimizzazione topologica, stampata e ripulita dai supporti di stampa
Structure resulting from topological optimization, printed and cleaned from print media
Winning model (right) and two challengers

Click to see the 3D render