Application Designing Additive Structures - ScienceDirect

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ScienceDirect Procedia Engineering 185 (2017) 131 – 138

6th Russian-German Conference on Electric Propulsion and Their Application Designing additive structures

Valeriy A. Komarova*, Alexander A. Pavlova, Svetlana A. Pavlovaa a

Aircraft Design Department, Samara University, Samara, Russian Federation

Abstract The objective of this paper is to demonstrate the opportunities of topology optimization applied to additive technology which will permit to design ultra-light structures practically without regard to the technological limits. The article gives a brief historical overview of the mutual influence of structures, materials and manufacturing technologies. The additive technology seems to have the broadest opportunities for producing existing structures using conventional materials without design changes. A hypothetical variable density material provides the means to solve an auxiliary problem of optimal material distribution considering stress or stiffness constraints. The special optimization algorithm allows the optimal topology layout to be found which will have a minimum value of the integral characteristic, called “load-carrying factor” (LCF). The LCF is a powerful tool for estimating the perfection limit of any structural and technological solution. Along with the optimal structure, such solutions are difficult for manufacturing using conventional materials and technologies. A creation of real material with the characteristics of hypothetical material is considered as one of the nearest areas of additive technology’s development for finding optimal structures. © 2017 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license © 2017 The Authors. Published by Elsevier B.V. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of RGCEP – 2016. Peer-review under responsibility of the scientific committee of the 6th Russian-German Conference on Electric Propulsion and Their Application Keywords: structure, additive technology, hypothetic material, optimization algorithm, load-carrying factor, optimal structure

1. Introduction Numerous objects of an artificial environment represent structures of various purposes. Many of them have the requirements for sufficient strength along with minimum weight. Throughout the entire development of civilization, design problems have been considered as part of the triad: material – technology – structure. The classification of structures – stony, metallic, riveting, welded, etc. – was established according to the importance and newness of one or two mentioned components (material and technology). A fundamentally new method for the creation of structures, additive technology, has appeared recently [1, 2]. The varied opportunities of this method and individual directions of its development promise a great technological effect [3]. The method provides the means to make structures from a wide variety of materials: polymers, different metals and all possible combinations of them [4]. It has become possible to produce monolithic structures with the internal cavities, truss structures, etc. * V.A. Komarov. Tel.: +7 846-267-46-45. E-mail address: [email protected]

1877-7058 © 2017 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the scientific committee of the 6th Russian-German Conference on Electric Propulsion and Their Application

doi:10.1016/j.proeng.2017.03.330

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An important task is to overcome the difficulties of using new technological opportunities the best way at design stages. First of all, the practical implementation of additive technologies began with the re-manufacturing of existing structures with complex shapes. This approach allows derivation from such complex conventional technologies as riveting, casting, welding and others. At the same time, the new technological opportunities presented by the additive method are not used to the full. The rapid development of additive technologies is not provided with a proper theoretical basics. It is appropriate to remind a well-known dialectic position: “Theory without practice is dead, but theory specifies the path to practice”. The opportunities of additive technologies allows to envisage the creation of ultra-light structures practically without any regard to the technological limits. Taking into account the theoretical point of view, it is interesting to know the perfection limit of a structure considering the given geometrical restrictions, bearing conditions and acting external forces. 2. Structural Design Methods In structural design, it is useful to distinguish two main types of activity: the choice of principle layout and rational dimensions of its elements. The first approach is called the “choice of load-carrying scheme” in the Russian literature. The second method is called “parametric optimization” under the influence of nonlinear mathematical programming. The load-carrying scheme determines the path of force transfer. Assigning rational dimensions of structural elements is aimed at ensuring strength and stiffness conditions with the minimum consumption of the material. During recent years, the first type of activity has become denoted by the term “topology optimization” under the influence of Western technical literature, the second type – “sizing”. Until now, the design of load-carrying structures has been characterized by a significant role of accumulated experience and engineering intuition. It especially concerns the choice of structural layout. The practice has developed two design principles, aimed at creating efficient structures. The first principle is the force transfer on the shortest paths. Sometimes it is very simple. For example, Fig.1a shows the transfer of concentrated load P to the sealing. The optimal solution is the rod which connects points A and B. If we change the load P direction to 90ι, the choice of optimal structure will be not so obvious (Fig. 1b).

a)

simple case

b)

complex case

Fig. 1. Different cases of load transfer

A more complex case is a three-dimensional task of choosing a structure of a low aspect wing. Fig. 2 represents two solution variants for the orientation of spars and stringers. Variant 2a shows the intuitive desire of transferring the air load by the shortest path. At the same time, the great efforts will appear in the caps, due to the low structural height of long rear spars. It is possible that variant 2b will be more advantageous in a weight ratio. It organizes the air load transfer through the area with a large structural height. In this case, the choice of optimal solution requires the use of special methods.

a)

Intuitive solution

b) Optimal solution Fig. 2. Optimal solutions for a low aspect wing

The second design approach represents the efficient action of the same structural elements in various calculated load cases. Moreover, this principle is essential for many load cases. With simplifications, the principle can be formulated as providing “mutual assistance” of elements to each other. The well-known Razani problem illustrates this approach [5].

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Fig.3 shows a truss loaded either by load P1 or load P2. A fully-stressed truss with two rods, 1 and 3, which is statically determined, have a greater mass than a statically indeterminate truss with rod 2. This rod works in both load cases and partly unloads rods 1 and 3 in each unit case. In practice, solution of such problems also requires the use of special optimization methods.

Fig. 3. Tree-rod truss

3. The most stiffness structures The article examines one of the possible approaches to solve the above described tasks. Following the main idea of A. Komarov (1965) [6], we will consider the auxiliary problem of designing structure with the minimum potential strain energy as the theoretical basis for building practical methods of solving such problems. New theoretical ideas of finding perfect structures can be efficient in a couple of additive technologies with almost unlimited capabilities. We assume that the structure is made of a limited by mass m amount of hypothetical material. This artificial material has an elastic modulus E, linearly depended on density ρ (1): E

U˜E ,

(1)

where E is the elastic modulus for a unit material density. A created 3D-model, defined by the boundary conditions of a structure, meshes with “solid” finite elements. An important feature of this approach is that all elements have the same density and Young’s modulus:

U 0i E0i

m , V

(2)

U 0i E ,

(3)

where subscripts denote: 0 – initial distribution, i – element number, V – total volume of all finite elements. The considered optimization problem can be stated as a minimization of the potential strain energy of a structure U o min

(4)

m const

(5)

subject to the mass constraint

In this case, the densities of individual finite elements are treated as design variables: xi

U i , i 1, ! n ,

6)

where n is the number of elements in the finite-element model. The variables U i are constrained considering mass conservation (7) n ¦ U iVi i 1

m,

where Vi – volume of element. The initial distribution of structural material density is calculated according to the equation:

(7)

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§ n · m ˜ ¨ ¦ Vi ¸ ¨ ¸ ©i 1 ¹

U oi

1

(8)

In the following stage, the principal stress within elements V10i , V 20i , V 30i is defined according to the energy failure theory: 2 V eqv

V 12  V 22  V 32  2P ˜ (V 1 ˜ V 2  V 2 ˜ V 3  V 3 ˜ V 1 )

The potential strain energy of a 3D-model could be written in the following form: 2 n V eqv o i U ˜ Vi ¦ i 1 2 Ui E

(9)

(10)

Equation (10) means that stresses of different elements are considered as constant, meanwhile densities U can i vary. In order to find the optimal values of the elements’ densities under proposed assumptions, we use the Lagrange multiplier method and write the following system of equations

wM ­ wU 0, (i 1,2,...,n) ° wU  O wU i ° i ® n °M ( U ) ¦ UiVi  m 0, i ° i 1 ¯

(11)

where O - Lagrange multiplier, M - auxiliary function, expressing the mass conservation (7). Taking into account (9), we differentiate the given system and obtain

­ V2 V ° eqv oi i   OVi ° ° 2 EU 2 i ® ° n ° ¦ U V m 0 °¯i 1 i i

(12)

0, (i 1,2,...,n)

This implies, that

Ui

V eqv oi

(13)

2 EO

Taking into account that mass conservation under density U i varies (7), the equation for new density distribution will be the following: (14) m , V eqv ij n V ¦ V eqvj i i 1 where U is a new density distribution, replacing initial density distribution U oi ; j – the iteration number. It should j 1 be mentioned than we assign new densities proportionally to the acting stresses. New densities lead to the new value of potential strain energy U10 described by stresses V eqv0i . Energy U10 can

U

j 1

be calculated by substituting U1i into equation (11). The initial potential energy U 0 of the 3D-model is characterized by stresses V (1, 2,3)0i and elements density U 0i . The new potential energy will be lower that the initial one, therefore

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U10  U 0 . At the same time, the 3D-model with density distribution U1i will have another stress distribution V (1, 2,3) ji , which would satisfy equilibrium and deformation consistency conditions. A finite-element model having stress distribution V (1,2,3) o i will satisfy only the equilibrium condition, but not the deformation ones. This means that the actual potential strain energy of a new structure with the density distribution U1i will become U1 . According to the principle of minimal strain energy we receive: (16) U1  U10 Re-calculation of the stress-strain state, using new density distribution U1i as the initial data, leads to a sequence of structures with potential energies

U 0 ! U1 ! U 2 ! ...

(17)

This monotonically decreasing sequence is bounded below because the strain energy is always positive. Thus the optimization process converges. The energy’s reduction to the minimal possible value means a search for the density distribution that gives a model with the maximum structural stiffness. Densities of some elements may tend to go zero during the optimization process. At the same time, the strain intensity becomes equal in non-degenerate elements. Let us consider the calculation of strain intensity at stage j+1 taking new densities into account (14): n

H j1

V eqvj1

V eqvj1 ˜ ¦ V eqvj ˜Vi

E j 1

m ˜ V eqvj ˜ E

(18)

i 1

When the stresses V eqvj1 become equal to V eqvj , the process converges, and we receive the following equation for strain: (19)

n

H j 1

¦ V eqvj ˜ Vi i 1

m˜ E

const

This property of the variable density 3D-body can be treated as strength balance. We can also prove (V. Komarov, 1999) that this iteration process leads to a structure with a minimal value of integral characteristics, which is called load-carrying factor (LCF). It simultaneously shows both the magnitude of the internal forces and the length of its path:

G

³V

eqv

˜ dV

(20)

V eqv

where V – equivalent stress, V – structural material volume. Subsequent iterations using (14) allow a structure with the most efficient way of external loads transfer over the prescribed volume V to be recieved. As a result, such a variable density structure can be considered as theoretically optimal. 4. Topology optimization algorithm We can formulate a topology optimization algorithm based on the considered auxiliary problem [7]. First of all, we need to complete property (1) with the similar strength condition:

>V @

U ˜ >V @ ,

(21)

In this case, the practical optimization algorithm for distributing hypothetic material takes the following form: 1. The specified geometric constraints are filled with an elastic continuous medium which is meshed with appropriate finite elements.

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2. Properties (2), (3) and (21) define the initial material distribution and its mechanical characteristics. These values are similar for unit finite elements. 3. Specific strength and elasticity are calculated. 4. Stress-stain state for density distribution U 0i is calculated. 2 V eqv

5.

V 12  V 22  V 32  2P ˜ (V 1 ˜ V 2  V 2 ˜ V 3  V 3 ˜ V 1 )

New densities and elastic modulus for each finite element are assigned: m , U V j 1 eqv ij n V ¦ V eqvj i i 1

E j 1i

U j 1i ˜ E ,

(22)

(22) (23)

where the first subscript is an iteration number, and the second one is the element number. 6. New values are assigned to the finite elements and the process is repeated until convergence. Convergence is usually reached after 15-20 iterations. 7. Rationality of the final load-carrying scheme is estimated using load-carrying factor. 8. The most advantageous structure is developed within strict technological and manufacturing constraints. 5. Practical application Let us consider an example of practical application of topology optimization algorithm using variable density idea. The task is stated as finding theoretically optimal three-dimensional structure loaded by torsion moment. The initial body represents a rectangular parallelepiped bounded by nodes A, B, C, D, E, F, G, H. The structural sizes are AB = 100 mm, BC = 500 mm, AE = 1000 mm. The load is applied at nodes A and C of the free end in the form of a pair of concentrated forces P

10000 N . The opposite side of the body is fixed.

The described optimization algorithm 1-8 leads to the optimal density distribution illustrated in Fig.4. For the results presented in the article, the darkest area corresponds to the maximum stiffness, and the lighter areas stand for lower stiffness.

Fig. 4. Optimal material distribution

Fig. 5 shows the optimal material distribution for a number of cross-sections. The whole structure can be divided into three typical regions: sealing area, the middle part and torque application area.

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a)

middle area

b) sealing area Fig. 5. Typical cross-sections

c)

tip area

The middle part (Fig. 5a) works almost as a thin-wall structure with a closed loop. Attention is drawn to a big area where densities tend to go zero inside the structure. Contour fillet is also formed by zero thickness at the corner elements. This part entirely corresponds to the intuitive consideration about the rational load-carrying scheme for transferring torsion moment. We can observe the transformation of the tubular part into the original beam structure on a short stretch near the sealing (Fig. 5b). Here, the applied torque is perceived by large tangential forces in the wall in the most efficient way. This area can be called “bimomented”. At the tip area (Fig. 5c), we can see a structure that converts a pair of concentrated forces in a closed flow of tangential forces, which are transferred further by the more efficient closed loop. The load-carrying factor of the theoretically optimal structure is equal to 1,16ή108 Nήmm. During the optimization process, 20 iterations were carried out. The reduction of load-carrying factor is about 12,5 per cent. The behavior of the load-carrying factor is shown in Fig. 6.

Fig. 6. Changing of load-carrying during iteration process

Another experiment was carried out to evaluate the opportunities of reliable designing of load-carrying structures. A beam was designed for a three-point bending test. The beam was made simultaneously with the samples for removing mechanical characteristics of material using a 3D-printer. Reliability was tested by comparing the failure stress of the beam with the failure stress in the samples. The beam stresses were calculated using finite-element method. Fig. 7 shows the beam during the test.

Fig. 7. Printed beam during the test

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In [8] application of the considered algorithm to the real problem of designing hinge fitting can be found. This task is characterized by forces acting on the unit in different planes. The Algorithm yielded the structure two times better than one designed according to the intuitive suggestions. The final structure has a rather complex configuration, and its manufacturing using additive technologies does not cause any problems. 6. Conclusion The considered topology optimization algorithm and solution of various tasks using this [9,10] shows that the approach allows rather perfect structures to be found, from the weight efficient point of view, with the peculiarity that, as a rule, they have variable density. It should be noted, that animal’s and especially bird’s bones are built in such way. Therefore, one of the tasks which engineers can put to the developers of additive technologies, is the creation of material having properties (1) and (2). Presumably, it could be structures, composed of tetrahedral with rods of different thickness, or hollow spheres with different diameters and thickness of walls. There are other possible solutions to create ultra-light structures. References [1] L.S. Baevs, A.A. Marinin, Modern technologies of additive manufacturing of objects, Vestnik MGTU, Vol. 17, No.1, 2014, pp.7-12 (in Russian). [2] Ian D. Harris, Development and implementation of metals additive manufacturing, Columbus 14 pp. [3] D.S. Thomas, S.W. Gilbert, Costs and cost effectiveness of additive manufacturing, NIST Special Publication 1176, 89 p. [4] http://software.materialise.com/cases/get-high-gear-altairs-optimized-race-car-brake-pedal [5] R. Razani, Behavoir of full-strength construction and its relation to the minimum weight structure, Rocket engineering and astronautics, 1965, No.2, pp. 35-39 (in Russian). [6] A.A. Komarov, Design fundamentals of load-carrying structures, Kuibyshev Book Publishers, 1965 (in Russian). [7] V.A. Komarov, Optimal design of aircraft load-carrying structures, doctoral thesis, Moscow aviation Institute, 1975 (in Russian). [8] V.A. Komarov, E.A. Kishov, R.V. Charkviani, Structural design and testing of Wing High-Lift Device, Journal of Machinery Manufacture and Reliability, 2016 Vol. 45, No. 5, pp. 476-483. [9] V.A. Komarov, A.V. Boldyrev, A.S. Kuznetsov, M.Yu. Lapteva, Aircraft design using variable density model, Aircraft Engineering and Aerospace Technology, Vo. 84, No. 3, 2012, pp. 162-171. [10] V.A. Komarov, A.V. Boldyrev, Application of the variable density model at the early stages of the wing design, TsAGI Science Journal, Vol. XLII, No.1, pp. 94-104 (in Russian).