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Recent Approaches for the Optimization of Lattice Structures. For the optimization of the design of lattice structures some few approaches have been. presented in the past Some of them deal with the optimization of the build up of frameworks. which is quite similar to the geometrical more filigree lattice structures and some deal with the. optimization of the densities of lattice structures The most important of the latter ones will be. introduced in the following section to illuminate their usability for flux of force adapted. structures, In 6 the influence of different parameters on the properties of lattice structures has been. investigated Therefore a torsion loaded hollow shaft has been build up of a helix shaped lattice. structure and different parameters like the angle of the helices relative to the torsion axis the. struts diameters and the number of struts in each direction have been varied The results have. been compared by the ratio of the shaft s torsion stiffness to its mass It has shown that the ratio. had the best values if the struts were orientated along the flux of force and if all of the struts. were loaded with the same stress In case of a torsion loaded hollow shaft it is quite easy to. reach this state because the stress conditions are almost equal all over the part However for. more complicated parts where the structure cannot be described by few parameters another. proceeding has to be developed, In 7 the most acknowledged calculation model for the strength of metallic cellular. materials has been developed The presented considerations are derived from assumptions of cell. deformation and failure Thus a first approach for the calculation of the compressive strength of. open cell lattice structures was derived It has shown that the presented approach is suitable until. relative densities up to 0 3, Rehme 8 carries out further investigations on the mechanical behavior of cellular. structures He expands the presented approaches in 7 for relative densities above 0 3. Furthermore he takes the anisotropic material behavior of additive manufactured solids and. lattice structures into account The investigations are of theoretical and experimental nature For. the latter ones the producibility of the structures by additive manufacturing processes is also. Both the approach of Gibson and Ashby 7 as well as the approach of Rehme 8 are. working with a homogenized material model of the in fact inhomogeneous lattice structures This. is feasible for the regarded periodic structures However for flux of force adapted structures with. varying densities as we can find them in the presented project this approach cannot be utilized. For this reason after a brief overview of the optimization procedure for flux of force. adapted lattice structures a necessary differing approach will be presented which regards the. structure as single struts, Procedure for Load dependent Optimization of Lattice Structures. In 9 a proceeding for the load dependent optimization of lattice structures has been. introduced Thereby the optimization regards two main aspects On the one hand the course of. lattice structures is getting adapted to the flux of force inside the respective part This has the. effect that bending loads on the single struts can be avoided or at least strongly reduced which. has a positive effect on the part s properties concerning lightweight design On the other hand. the geometries of the single struts and the nodes in the structure are optimized with respect to. their loads The general proceeding for these optimizations can be seen in Fig 1. Process specific constraints,Course of struts,Postprocessing.

Preprocessing,Calculation of loads,Determination of strut and node geometries. Fig 1 Proceeding for the load adapted optimization of lattice structures according to 9. The optimization is realized with the software NX by Siemens and its integrated. NASTRAN solver Therefore several algorithms were written in MATLAB and NXOpen which. is the corresponding programming language for NX Anyway the optimization can also be done. with other CAE systems and calculation programs However for the presented project NX has. been chosen because it provides all of the needed CAE tools and it is comprehensively. programmable with NXOpen, The optimization process begins with the preprocessing of the optimization data Here. the design space for the optimization as well as component parts which will not be optimized in. the later optimization procedure e g for force application screw treads or bearing carriers are. modeled in a CAD system For this geometry a finite elements model is generated which. contains all of the constraints and loads for the later structure The model is solved and the solver. output data is used for the following build up of the structure. The optimization is executed under consideration of the process specific constraints. which arise from the used additive manufacturing process Hereunto numerous investigations. for the material AlSi12 have been done in the past Therefore extensive tensile tests have been. carries out in order to determine the dependencies of the manufacturability of the single struts. and their mechanical properties from several parameters like build up angle strut length and. strut diameter The results have been integrated in an anisotropic geometry dependent material. model which describes these correlations This model is used in the following optimization. process for the orientation of the structure as well as the optimization of the struts diameters 10. The optimization process starts with the flux of force dependent build up of a basic. structure Therefore the flux of force in the design space has to be determined which is executed. based on the solver data from the preprocessing For this purpose the main stress directions for. each element respectively node is calculated With the help of these vectors a sequence of points. can be determined which represent the flux of force respectively the main stress curves The. intersection points of these curves characterize the nodes for the later lattice structure When all. of these nodes have been constructed straight beams are located between the respective. correlating nodes compare Fig 3 11 Unfortunately this build up process of the three. dimensional structure from the lines for the flux of force still needs high manual effort. Therefore in future works an automated algorithm will be developed which executes this task. and reduces human interaction, For the resulting structure suitable diameters for the struts have to be found which fulfill. predefined requirements like maximum stresses or minimum diameters The proceeding for this. optimization is the main content of this contribution and will be described in the following. sections In addition to that the material model which was mentioned before will be integrated. into the optimization of the diameters in order to incorporate the anisotropic geometry. dependent material behavior in the optimization, Besides the geometry of the struts the design of the single nodes has to be optimized too. A smooth merging between the single struts in the nodes has to be realized to avoid high notch. stresses see Fig 13 A suitable design of these nodes as well as an automated algorithm to. realize them in 3D CAD systems respectively finite elements systems here Siemens NX will be. developed in future works, In the postprocessing the optimized structure has to be made applicable for subsequent.

processes Therefore a 3D CAD model can be build up for further design steps or data. processing for the following manufacturing of the optimized structure A respective NXOpen. program has been developed in the past to perform this task. Results for the Optimization of the Course of Lattice Structures. In 5 lattice structures which are adapted to the flux of force inside a respective part. are compared to a structure with a regular build up For that purpose a design space with. constraints and forces as depicted in Fig 2 is used as initial situation for the respective. optimizations,fixed bearing,shearing load, Fig 2 Design space constraints and load for optimization according to 5. The material used for the investigations is aluminium 6061 without thermal treatment. which has the following mechanical properties,Young s modulus 58 1 GPa. Limit of elasticity 55 MPa,Tensile strength 125 MPa. Density 2 711 g cm, The strut diameter for all structures is kept constant at 2 mm because only an adaption of. the course of the lattice structure should be considered. In the investigations the maximum von Mises comparison stress is determined in order. to rate the stability of the particular structures It is assumed that the maximum stress linearly. depends on the applied force on the structure Hence the force at the limit of elasticity can be. calculated The ratio of this load to the respective mass is used to compare the different. structures whereat a higher ratio indicates a better lightweight design. Initially a periodic structure build up is investigated This is the currently established. approach for the application of lattice structures in lightweight design The structure has a total. mass of 147 5 g The corresponding maximum value for the von Mises stress is 50 58 N mm. Under the assumption that this maximum stress linearly depends on the applied force a. maximum load of 406 6 N can be calculated for the obtaining of the limit of elasticity The ratio. of this force to the part s mass is 2 76 N g This value is used in order to compare the structure. with the other ones in the paper, Another structure which is presented in 5 is adapted to the flux of force in the design.

space see Fig 2 Here the single nodes of the structure are located along the fluxes of force in. main stress direction 1 2 and 3 These nodes are connected by straight struts to build up a. structure This build up has the advantage that almost no bending loads appear on the single. struts Analogue to the investigations presented before the diameters of the struts is set constant. The resulting structure and its constraints and loads can be seen in Fig 3 The mass of. this geometry is 83 58 g Thus the weight is 43 less than with conventional periodic. structures, Fig 3 Shear loaded beam with flux of force adapted lattice structure 5. The resulting maximum von Mises stress is 31 9 N mm Under the assumption that this. maximum stress linearly depends on the applied force a maximum load of 516 8 N can be. calculated for the obtaining of the limit of elasticity For the resulting ratio of this force to the. structure s mass a value of 6 18 N g can be calculated Hence the ratio has increased for 124. compared to the periodic structure, Thus it can be summarized that the adaption of the structure s course to the flux of force. in the design space has a great potential for lightweight design This is grounded in the fact that. almost no bending loads appear for this kind of structure 5. Uniform Optimization of Strut Diameters in Lattice Structures. The results of 5 which have been presented in the section before have been determined. for the arbitrary chosen diameter of 2 mm However to reach a predefined maximum von Mises. y 162 96x 2 003, stress or displacement additional investigations have to be done In the following section such. maximumcalculations,von Mises stress, are presented for a beam which is similar to the one presented before. For this purpose the structure is built up with 1D CBEAM elements and the strut. diameters are varied uniformly for the whole structure The influence of these strut diameters on. the mechanical properties in this case the maximum von Mises stress is determined The results. of these investigations can be seen in Fig 4,4545 maximum.

maximum von Mises stress,von Mises stress,maximum von Mises stress. strut diameter Stabdurchmesser mm, Fig 4 Maximum von Mises stress in dependence of the struts diameter. Because almost solely push and pull forces appear on the struts the maximum von Mises. stress is indirectly proportional to the cross sectional area and thereby indirectly proportional to. the squared diameter, For this and the following example the optimization goal is set to a maximum von Mises. stress of 32 N mm This result is used for strut diameters of 2 26 mm which means a mass of. 104 g for the structure In combination with the maximum von Mises stress of 31 8 N mm and. the force at the limit of elasticity of 518 4 N a ratio of force to mass of 4 97 N g can be. calculated This structure will also be the basis for the optimization of the single strut diameters. in the following section, Optimization of Strut Diameters in Lattice Structures. Beyond the optimizations of the course of lattice structures and the uniform optimization. of the struts diameter the cross section of the single struts can also be varied independently of. each other Thereby a constant material saturation can be reached within the part which leads to. an enhanced lightweight design The computer based proceeding for this optimization can be. seen in Fig 5,check if all struts meet,the predefined criteria.

buildup of structure,end of optimization,optimization yes. Fig 5 Proceeding for the optimization of the strut diameters in lattice structures. The optimization starts with the automated build up of the initial structure by a macro for. the CAE software NX from Siemens Therefore the nodes are placed along the flux of force as. described before These points are connected with 1D CBEAM elements which are described by. the corresponding nodes and the respective diameters of the circular cross sections Furthermore. the occurring constraints and loads are applied on the structure. When the initial structure is completed the actual optimization circle starts Here the. van Mises stresses in the struts of the initial structure are calculated Subsequently the algorithm. checks if all struts meet the predefined exit conditions e g designated stress minimum. diameter In the case that this is true the procedure stops and the optimization has finished If. not the struts diameters are adapted in dependence of the designated stress and the respective. appearing stresses Therefore different functions can be applied which has an influence on the. optimization time For the resulting structure the stresses are calculated again and the cycle. starts over until all struts meet the exit conditions. This optimization procedure has been applied to the structure which has been presented. before for a shear loaded beam see Fig 3 For this purpose the structure has been build up with. CBEAM elements which have fixed connections among each other in the respective nodes For. the optimization a target von Mises stress of 31 32 N mm as well as a minimum strut diameter. of 0 5 mm have been set as exit condition The resulting structure can be seen in Fig 6. Fig 6 Structure which was optimized with fixed nodes and a maximum von Mises stress as optimization goal. The result is a very inhomogeneous structure There are local areas with very thick struts. while the there is almost no material behind these areas The reason for this behavior can be. found in inappropriate stress states while the optimization cycle Here bending loads can appear. on single struts due to the deformation of the loaded structure which leads to stress peaks Since. the new diameters are calculated on the basis of this maximum stress these struts are growing. disproportionately high This leads to reduced loads on the struts behind these massive areas so. that these struts are getting more resilient Thus the thick beams are loaded even more As a. result the thick struts are getting thicker until they reach the maximum acceptable von Mises. stress At the same time the thin struts are getting thinner until they reach the minimum. acceptable diameter, The resulting distribution of the von Mises stress in the structure can be seen in Fig 7. The structure has a mass of 106 g and a maximum stress of 31 38 N mm The force at which. the limit of elasticity is reached for this structure is 525 8 N From these values a characteristic. coefficient of 4 960 N g can be calculated for the ratio of load capacity to mass This even means. a minor worsening compared to the initial structure The reason for that can be found in the. increased appearance of bending loads on single struts due to the unfavorable diameters. Von Mises stress,Units N mm, Fig 7 Von Mises stress in a structure which was optimized with fixed nodes and constraints and a maximum. von Mises stress as optimization goal, Furthermore it can be seen that the distribution of the stress over the structure is very. inhomogeneous Ideally most of the struts should show a similar stress near the designated value. here 31 0 31 9 N mm, Therefore an alternative approach for the optimization of the diameters has to be found.

which is not negatively influenced by the bending loads on the single struts while the. optimization cycle To reach a stress state which does not show any bending loads inside the. structure the rotational degrees of freedom DOF 4 5 and 6 at the connection nodes between. the beams are released as well as the respective degrees of freedom in the constraints Thus. moments can no longer be transmitted which avoids the appearance of bending loads. This proceeding is feasible for a first optimization step because ideally there should not. appear any bending forces in the flux of force optimized structure anyway However a. subsequent calculation of the strut loads with fixed DOFs in the constraints and nodes is. necessary after the optimization The results for the optimized structure can be seen in Fig 8. As it can be recognized the structure shows a way smoother distribution of the struts. diameters Here the forces are transmitted along the strut courses from the force application. points to the constraints This leads to a reduced mass of 58 1 g for the structure. Fig 8 Structure which was optimized with released rotational degrees of freedom at the nodes and constraint. and a maximum von Mises stress as optimization goal. The resulting von Mises stresses for this structure can be seen in Fig 9. Von Mises stress,Units N mm, Fig 9 Von Mises stress in a structure which was optimized with released nodes and constraints and a. maximum von Mises stress as optimization goal, As it can be seen the resulting distribution of the stress over the structure is way more. homogeneous than for the optimization presented before which indicates a good lightweight. design Most of the struts meet the predetermined stress values The remaining struts are. restricted by the minimum diameter of their cross section. Hence a maximum von Mises stress of 31 94 N mm appears in the structure The force. at which the limit of elasticity is reached for this structure is 516 6 N From this value a. characteristic coefficient of 8 891 N g can be calculated for the ratio of load capacity to mass. This means an enhancement of 79 compared to the structure presented before and to the initial. structure The reason for that can be found in the strongly reduced appearance of bending loads. on single struts and the way more uniform distribution of the stress in the structure. However this proceeding leads to a problem concerning the automated optimization of. the struts diameters Since the system is underdetermined if all rotational degrees of freedom are. released single minor loaded DOFs have been fixed manually to reach a statically determined. state of the structure Otherwise the solver cannot calculate the model and the corresponding. finite elements software produces an error However because the optimization procedure is. meant to work automated for complex structures this manual interaction is not suitable. For this reason an alternative approach for the optimization has been developed which. also is not negatively influenced by bending loads on the single struts but in contrast to the. preceding approach does not need any manual interaction. To explain this alternative proceeding the stress conditions in beam elements for the used. NASTRAN solver have to be explained For beam elements the calculation of the stresses is. executed at four characteristic points the so called stress recovery points C D E and F see Fig. C cross section of,a beam element,stress recovery point. Fig 10 Stress recovery points C D E and F at the ends of beam elements for a NASTRAN solver. The push respectively pull part of the combined load in a beam element can be. determined out of the points stresses by calculating their average value If this average stress is. used for the optimization of the struts diameters instead of the maximum value the bending. loads do no longer have an influence on the result Hence an optimized structure as presented. before can be achieved without manual interaction The result from this optimization approach. can be seen in Fig 11, Fig 11 Structure which was optimized with fixed rotational degrees of freedom at the nodes and constraints. and an average von Mises stress as optimization goal. As before the structure shows a smooth distribution of the struts diameters This. approach leads to a comparable mass of 57 5 g The maximum von Mises stress is comparable as. well with a value of 31 95 N mm see Fig 12,Von Mises stress.

Units N mm, Fig 12 Von Mises stress in a structure which was optimized with fixed nodes and constraints and an average. von Mises stress as optimization goal, Hence a proceeding for optimization of the struts diameters has been developed which. works without manual interaction for flux of force adapted structures. In structures which are not adapted to the flux of force however bending loads do appear. in the struts Therefore the last two proceedings do not work Here the first procedure has to be. adopted The optimization results for the investigated structures are summarized in Table 1. von Mises stress force at limit of maximum force,N mm elasticity N mass N g. 104 31 83 518 5 4 97,fixed nodes,106 31 38 525 8 4 96. maximum stress,released rotational,58 1 31 94 516 6 8 891.

DOFs maximum stress,fixed nodes,57 5 31 95 516 4 8 981. average stress, Table 1 Summary of the properties of the investigated structures. In Fig 13 a 3D simulation of the structure can be seen which has been optimized with. fixed nodes and constraints and an average von Mises stress as optimization goal Comparable to. Fig 12 with beam elements it is recognizable that the stresses in the struts are almost equal all. over the structure, Fig 13 Simulation of the optimized structure with 3D elements. However there do appear notch stresses in the nodes where the struts are connected. Therefore in future works there has to be found a better design for the nodes where the struts. merge smoother and therefore no notch stresses appear. Summary and Outlook, It has been exemplified that Additive Layer Manufacturing has a great potential for the. production of lightweight components Especially mesoscopic approaches like lattice structures. exhibit great properties for mass reduction However these structures are currently designed as. periodic patterns This leads to unfavorable bending loads in the single struts and uneven stresses. over the structure, To improve the structures build up an optimization approach has been introduced which.

adapts the course of the structure to the flux of force in a part The great potential of this. proceeding has been shown at the example of a shear loaded beam Here it was possible to. enhance the ratio of the maximum applicable force to the structures mass for 124 compared to. a periodic structure, Another optimization goal of the presented approach is the adaption of the struts. diameters with respect to the appearing stresses First an optimization has been introduced at. which the struts diameters are optimized uniformly until a predetermined maximum stress is. reached Thus it was possible to almost exactly reach the desired stress of 32 N mm However. the loads in the single struts were still varying, Therefore another second optimization approach has been applied Here the single struts. were adapted to their respective stresses individually However several different procedures. concerning the fixed and loose rotational degrees of freedom and the optimization goal. maximum of average von Mises stress had to be investigated to come to an optimization. procedure which does not lead to unwanted bending loads in the beams and which can be. performed without any manual interaction Thus the ratio of the maximum applicable force to. the structures mass could be enhanced for 80, However FEM calculations with a tree dimensional model of the structure have shown. that there are severe notch stresses appearing at the nodes where the beams are merging. Therefore a suitable node design will be developed in future works which realizes a smooth. connection of the struts and therefore reduces stress peaks. Beyond that another procedure for the optimization of the struts diameters will be. developed which makes it possible to reach a predefined stiffness of the structure with the. application of a minimum mass, Furthermore a rule based algorithm will be developed which is able to automatically. build up a structure that is adapted to the flux of force of a part Currently this step has to be. done with great manual effort, The presented approach for the optimization of lattice structures has been applied on the.

example of a shear loaded beam However further optimizations will be done on real. components from industry in the future Thereby the most benefit can be reached for parts which. are strongly accelerated or in industries where lightweight design is of essential importance. Therefore possible applications can be found in aviation and space flight in automotive. industries especially electric mobility and racing as well as in production and processing. Acknowledgements, We want to express our gratitude to the state government of Bavaria for its financial. support in the build up of the Fraunhofer Project Group for Resource Efficient Mechatronic. Processing Machines RMV Thereby the research work on the topic presented in this paper. was enabled,References, 1 Wohlers T 2010 Wohlers Reprt 2010 Additive Manufacturing State of the Industry. Wohlers Assiciates Inc Fort Collins 2010, 2 Zaeh M F Ott M 2011 Investigations on heat regulation of additive manufacturing. processes for metal structures CIRP Annals Manufacturing Technology 60 2011. 3 Kruth J P Levy G Klocke F Childs T H C 2007 Consolidation phenomena in. laser and powder based layered manufacturing Annals of the CIRP Vol 56 2 2007. 4 Reinhart G Teufelhart S Ott M Schilp J 2010 Potentials of Generative. Manufactured Components for Gaining Resource Efficiency of Production Facilities In. Neugebauer R Ed International Chemnitz Manufacturin Colloquium Reports form. the IWU Volume 54 703 710 Verlag Wissenschaftliche Scripten Auerbach 2010. 5 Teufelhart S 2012 Investigation of the Capability of Flux of Force Oriented Structures. for Lightweight Design Proceedings of the 2nd WGP Jahreskonferenz June 26th June. 27th 2012 Berlin Germany, 6 Reinhart G Teufelhart S 2011 Load Adapted Design of Generative Manufactured. Lattice Structures Physics Procedia 12 2011 pp 385 392. 7 Gibson L J Ashby M F 1997 Cellular Solids Structure Properties Second. Edition Cambridge University Press Cambridge 1997, 8 Rehme O 2010 Cellular Design for Laser Freeform Fabrication Cuvillier Verlag.

G ttingen 2010, 9 Reinhart G Teufelhart S 2011 Approach for Load adapted Optimization of. Generative Manufactured Lattice Structures In Spath D Ilg R Krause T Eds ICPR21. 21st International Conference on Production Research Conference Proceedings. 10 Reinhart G Teufelhart S Riss F 2012 Examination of the Geometry dependent. Anisotropic Material Behavior in Additive Layer Manufacturing for the Calculation of. Mesoscopic Lightweight Structures Proceedings of the Fraunhofer Direct Digital. Manufacturing Conference 2012 March 14th March 15th 2012 Berlin Germany. 11 Reinhart G Teufelhart S 2012 Optimization of Mechanical Loaded Lattice. Structures by Orientating their Struts Along the Flux of Force Proceedings of the 8th. CIRP Conference on Intelligent Computation in Manufacturing Engineering July 18th.

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