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Table 1. Cont. After the parameters are defined in the skeleton model, they are published and used as external parameters during the phase of the definition of the shape of external fixation device, and during the definition of position for some of the components of the external fixation device. Relationships inside the skeleton model give flexibi change the shape, place, and orientation of external fixation device ity to the design to components. Rela- tionships reference design parameters and geometrical elements, which are used to place the components of the external fixation device with correlation to t he main coordinate system of skeleton model. Except parameters and relationships, the skeleton model was inside all other referent elements, such as points, planes, and axes (Figure 2). The places of all elements are defined in correlation to the main coordinate system of the skeleton model using point coordinates or other dimensions. The local coordinate system of all components correlates with the main coordinate system of the skeleton model. Because of this, the skeleton model gives full flexibility to the external fixation dev ice design. It is easy to change the position or geometry of some components. This is a basic condition to carry out the structural optimization process.

Table 1 Cont. After the parameters are defined in the skeleton model, they are published and used as external parameters during the phase of the definition of the shape of external fixation device, and during the definition of position for some of the components of the external fixation device. Relationships inside the skeleton model give flexibi change the shape, place, and orientation of external fixation device ity to the design to components. Rela- tionships reference design parameters and geometrical elements, which are used to place the components of the external fixation device with correlation to t he main coordinate system of skeleton model. Except parameters and relationships, the skeleton model was inside all other referent elements, such as points, planes, and axes (Figure 2). The places of all elements are defined in correlation to the main coordinate system of the skeleton model using point coordinates or other dimensions. The local coordinate system of all components correlates with the main coordinate system of the skeleton model. Because of this, the skeleton model gives full flexibility to the external fixation dev ice design. It is easy to change the position or geometry of some components. This is a basic condition to carry out the structural optimization process.

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eG aaa, CD ARR, AE CAO anaes a aie KBE systems and structural optimization are widely used for the development and yptimization of fixation devices and implants [12-14]. External fixation device is a medica Jevice which is used to immobilize bone fraction using pins, which goes inside and through one, and which is externally connected to external mounting. There are a lot of design, biomechanical and manufacturing constrains which must be fulfilled in the design of external fixation device. The most important constrains are the safety of usage and practica 1pplication to the bone. The external fixation device, which is analyzed in this paper, is a inilateral, biplanar external modular fixation device with half pins (Schanz screws) [15,16]. [he concept of developed KBE system for structural dimensional optimization of externa ‘ixation device is shown in Figure 1. Using sensors inside structural analysis, displacements are measured and imported into an optimization model as constrains. In this way, displacements and stress are con- trolled on important places on an external fixation device. The data of design parameters are imported into the CAD model af ter every iteration of the optimization process, and then forwarded to the optimization model. Moreover, the other design parameters that are needed to calculate the volume us into the optimization model. Using ed material for external fixation device are imported these parameters, volume (as cost function) can be calculated. The described process represents one iteration of the developed KBE system. The iteration process is repeated until sults are obtained. The initial CAD/F one of the constrains is reached, and converges of re- EM model is verified using experimental verification on the real design of the external fixa tion device (Figure 1).
eG aaa, CD ARR, AE CAO anaes a aie KBE systems and structural optimization are widely used for the development and yptimization of fixation devices and implants [12-14]. External fixation device is a medica Jevice which is used to immobilize bone fraction using pins, which goes inside and through one, and which is externally connected to external mounting. There are a lot of design, biomechanical and manufacturing constrains which must be fulfilled in the design of external fixation device. The most important constrains are the safety of usage and practica 1pplication to the bone. The external fixation device, which is analyzed in this paper, is a inilateral, biplanar external modular fixation device with half pins (Schanz screws) [15,16]. [he concept of developed KBE system for structural dimensional optimization of externa ‘ixation device is shown in Figure 1. Using sensors inside structural analysis, displacements are measured and imported into an optimization model as constrains. In this way, displacements and stress are con- trolled on important places on an external fixation device. The data of design parameters are imported into the CAD model af ter every iteration of the optimization process, and then forwarded to the optimization model. Moreover, the other design parameters that are needed to calculate the volume us into the optimization model. Using ed material for external fixation device are imported these parameters, volume (as cost function) can be calculated. The described process represents one iteration of the developed KBE system. The iteration process is repeated until sults are obtained. The initial CAD/F one of the constrains is reached, and converges of re- EM model is verified using experimental verification on the real design of the external fixa tion device (Figure 1).
Figure 2. Parametric skeleton model (a) and CAD model of external fixation device (b). The so-called top-down method is used to develop the generative CAD model. This method means that parameters and relations are used to develop a generative model. n addition, this method implies that the so-called skeleton model of the external fixa- tion device needs to be developed. The skeleton model represents the infrastructure of the external fixation device; also, it defines relations between the parameters of design Figure 2). Knowledge about the design of, and relationships between, design components is integrated into the skeleton model. This represents a basis of so-called generative model- ing. The generative model is not the same as the parametric model; the generative model forms the knowledge base about different designs of external fixation device in a form of different CAD models. Moreover, the knowledge base can hold other information from other sources; for example, from the experimental testing of the real design of the external fixation device. The skeleton model must be developed first in the process of the development of the generative CAD model; information about the design and relationships between the components of design must be known (Figure 2). Moreover, all of the other parameters that are needed for the function of the external fixation device must be imported into the skeleton model [17].
Figure 2. Parametric skeleton model (a) and CAD model of external fixation device (b). The so-called top-down method is used to develop the generative CAD model. This method means that parameters and relations are used to develop a generative model. n addition, this method implies that the so-called skeleton model of the external fixa- tion device needs to be developed. The skeleton model represents the infrastructure of the external fixation device; also, it defines relations between the parameters of design Figure 2). Knowledge about the design of, and relationships between, design components is integrated into the skeleton model. This represents a basis of so-called generative model- ing. The generative model is not the same as the parametric model; the generative model forms the knowledge base about different designs of external fixation device in a form of different CAD models. Moreover, the knowledge base can hold other information from other sources; for example, from the experimental testing of the real design of the external fixation device. The skeleton model must be developed first in the process of the development of the generative CAD model; information about the design and relationships between the components of design must be known (Figure 2). Moreover, all of the other parameters that are needed for the function of the external fixation device must be imported into the skeleton model [17].
Table 1. External parameters of sub algorithms for components design of external fixation device.
Table 1. External parameters of sub algorithms for components design of external fixation device.
Defined properties of orthotropic linear elastic material must fulfill following conditions
Defined properties of orthotropic linear elastic material must fulfill following conditions
Figure 4. FEM model of external fixation device. Table 2. Mechanical properties of a beech tree.
Figure 4. FEM model of external fixation device. Table 2. Mechanical properties of a beech tree.
Figure 5. Non-deformed and deformed structures of the fixator under maximum axial load and interfragmentary displace- ments at the fracture. FEM model of external fixation device before, and after applying loads for axia pressure are given on Figure 5, together with interfragmentary displacements.
Figure 5. Non-deformed and deformed structures of the fixator under maximum axial load and interfragmentary displace- ments at the fracture. FEM model of external fixation device before, and after applying loads for axia pressure are given on Figure 5, together with interfragmentary displacements.
Table 3. Values of displacements under maximum intensity of load.
Table 3. Values of displacements under maximum intensity of load.
Figure 6. Principal stresses under maximum axial load. The full numerical structural analysis of external fixation device, except the analysis of displacements at fracture gap, includes also the analysis of principal stress at characteristic places of design [14,20]. During FEM and experimental analysis, values and directions of principal stress are calculated at two control places at the middle part of the fixator tree. The place which is closer to the bone segment model is marked with MP-, and the place on another side is marked with MP+ (Figure 6, detail A).
Figure 6. Principal stresses under maximum axial load. The full numerical structural analysis of external fixation device, except the analysis of displacements at fracture gap, includes also the analysis of principal stress at characteristic places of design [14,20]. During FEM and experimental analysis, values and directions of principal stress are calculated at two control places at the middle part of the fixator tree. The place which is closer to the bone segment model is marked with MP-, and the place on another side is marked with MP+ (Figure 6, detail A).
Table 4. Maximum values of principal and von Mises stresses at the measuring points. 4. Experimental Analysis Pressure stress is detected at MP— place, and they have bigger intensity in correspon: dence with tensile stress at MP+ place. This is because the pressure in the tree of external fixation device is eccentric. The direction of the biggest principal stress 0, at place MP+ is the same as the direction of the lowest principal stress 03 at place MP—, coinciding with the axis of symmetry of the tree of the external fixation device. Direction and intensity of principal stresses at measuring places are shown Figure 6 (view B). It can be seen that at place MP+, the biggest principal stress is tensile stress, and at place MP—, the smaller principal stress is pressure stress. In additional, it can be noticed that principal stresses (oj and o3) are in the bending plane of the fixator. This plane does not match with the plane of half pins (Figure 6, view B). This is because of eccentric pressure load and the geometry of the external fixation device. The intensities of three principal stresses are shown in Table 4 Today, because of the height level of technology development, the aim is to achieve full utilization of the material for the manufacturing of the external fixation device, be- cause of that, working loads are pushed to their maximum. The first step is to do an analysis about stress distribution using experimental analysis. The next step is to carry out the optimization process to achieve lower mass in correspondence to allowed interfrag- mentary displacement and stresses at defined places of design.
Table 4. Maximum values of principal and von Mises stresses at the measuring points. 4. Experimental Analysis Pressure stress is detected at MP— place, and they have bigger intensity in correspon: dence with tensile stress at MP+ place. This is because the pressure in the tree of external fixation device is eccentric. The direction of the biggest principal stress 0, at place MP+ is the same as the direction of the lowest principal stress 03 at place MP—, coinciding with the axis of symmetry of the tree of the external fixation device. Direction and intensity of principal stresses at measuring places are shown Figure 6 (view B). It can be seen that at place MP+, the biggest principal stress is tensile stress, and at place MP—, the smaller principal stress is pressure stress. In additional, it can be noticed that principal stresses (oj and o3) are in the bending plane of the fixator. This plane does not match with the plane of half pins (Figure 6, view B). This is because of eccentric pressure load and the geometry of the external fixation device. The intensities of three principal stresses are shown in Table 4 Today, because of the height level of technology development, the aim is to achieve full utilization of the material for the manufacturing of the external fixation device, be- cause of that, working loads are pushed to their maximum. The first step is to do an analysis about stress distribution using experimental analysis. The next step is to carry out the optimization process to achieve lower mass in correspondence to allowed interfrag- mentary displacement and stresses at defined places of design.
Figure 7. Experimental analysis of external fixation device for axial pressure load.
Figure 7. Experimental analysis of external fixation device for axial pressure load.
Figure 8. Placement of strain gauges on fixator tree. In this case, bending deformations are much larger than pressure deformations (|es] >> €p). According to that, this case of load has unequally distributed principal deformations along the longitudinal section of the tree. The neutral line does not match the axis of symmetry of the fixator tree (Figure 8). Because of this, two Wheatstone quar- ter bridges are applied and connected to the Quantum X system using two measuring channels. Wheatstone quarter bridges have one active SG and one compensation (passive)
Figure 8. Placement of strain gauges on fixator tree. In this case, bending deformations are much larger than pressure deformations (|es] >> €p). According to that, this case of load has unequally distributed principal deformations along the longitudinal section of the tree. The neutral line does not match the axis of symmetry of the fixator tree (Figure 8). Because of this, two Wheatstone quar- ter bridges are applied and connected to the Quantum X system using two measuring channels. Wheatstone quarter bridges have one active SG and one compensation (passive)
Figure 9. Wheatstone quarter bridge with active SG and compensation SG2 strain gauge strain gauge SG2 (Figures 8 and 9) of the same type. Compensation strain gauge is ap- plied on the small, unloaded part, which is connected to fixator tree close to active strain gauge (Figures 7 and 8). The unloaded part is manufactured from the same material as the fixator tree. Using basic equation of the Wheatstone bridge:
Figure 9. Wheatstone quarter bridge with active SG and compensation SG2 strain gauge strain gauge SG2 (Figures 8 and 9) of the same type. Compensation strain gauge is ap- plied on the small, unloaded part, which is connected to fixator tree close to active strain gauge (Figures 7 and 8). The unloaded part is manufactured from the same material as the fixator tree. Using basic equation of the Wheatstone bridge:
Figure 10. Comparative diagram of the principal stresses (0, on MP+) and (03 on MP—),
Figure 10. Comparative diagram of the principal stresses (0, on MP+) and (03 on MP—),
Figure 11. Comparative diagram of the axial displacement d at the point of load.
Figure 11. Comparative diagram of the axial displacement d at the point of load.
Table 5. Analogy between real physics annealing and SA method for optimization. NE sO It can be noticed that, at higher temperatures, the system has almost the same pos- sibility to reach any energy level, but at lower temperatures, the system has the small possibility of reaching height energy level (8). Table 5 shows an analogy between real physics annealing and the SA method for optimization.
Table 5. Analogy between real physics annealing and SA method for optimization. NE sO It can be noticed that, at higher temperatures, the system has almost the same pos- sibility to reach any energy level, but at lower temperatures, the system has the small possibility of reaching height energy level (8). Table 5 shows an analogy between real physics annealing and the SA method for optimization.
Table 6. Optimization parameters (referent design parameters) of external fixation device. where: x is vector of reference construction parameters which are used for fixator parametriza- tion (Table 6), AyL; is the volume of fixator elements, and i is the element of design. External fixation device has 61 elements in total.
Table 6. Optimization parameters (referent design parameters) of external fixation device. where: x is vector of reference construction parameters which are used for fixator parametriza- tion (Table 6), AyL; is the volume of fixator elements, and i is the element of design. External fixation device has 61 elements in total.
Figure 12. Optimization constrains on FEM model. where: 6,1 is maximal resulted displacement of proximal bone segment at fracture gap, and 5g = 1—2mm is recommended for the maximal displacements of proximal bone segment at fracture gap for given fracture gap size [27].
Figure 12. Optimization constrains on FEM model. where: 6,1 is maximal resulted displacement of proximal bone segment at fracture gap, and 5g = 1—2mm is recommended for the maximal displacements of proximal bone segment at fracture gap for given fracture gap size [27].
Figure 13. Algorithm of developed KBE system.
Figure 13. Algorithm of developed KBE system.
Figure 14. Optimization of volume of external fixation device.
Figure 14. Optimization of volume of external fixation device.
Table 7. Comparison between initial and optimized design of external fixation device. The SA algorithm, after 33 iterations, completely converges to the best solution of cost function (Figure 14), and achieves the optimal solution of analyzed design param- eters, which minimizes the value of the external fixation device volume (Figure 15). In addition, optimization constrains in the form of displacements and von Mises stresses are fulfilled (Figures 16 and 17). The CG algorithm did not have significant influence on the optimization parameters.
Table 7. Comparison between initial and optimized design of external fixation device. The SA algorithm, after 33 iterations, completely converges to the best solution of cost function (Figure 14), and achieves the optimal solution of analyzed design param- eters, which minimizes the value of the external fixation device volume (Figure 15). In addition, optimization constrains in the form of displacements and von Mises stresses are fulfilled (Figures 16 and 17). The CG algorithm did not have significant influence on the optimization parameters.
Figure 15. Changes of optimization parameters.
Figure 15. Changes of optimization parameters.
Figure 16. Reduction of displacement of proximal bone segment (Dp) during optimization process.
Figure 16. Reduction of displacement of proximal bone segment (Dp) during optimization process.
Figure 17. Difference between calculated and allowed von Mises stresses (Oym—) at fixator tree duri the optimization process.
Figure 17. Difference between calculated and allowed von Mises stresses (Oym—) at fixator tree duri the optimization process.
Figure 18. Comparison diagram of displacements at the place of fracture R and at the place of load for initial and optimized design. Significant decreases of displacement of bone segment Dp (68.31%) (Figure 16) and interfragmentary displacement Dg (R) (Figure 18) are achieved by increasing the diameter of the external fixation device tree (56.37%), which results in a significant increase of the resistance moment (108.52%), and a reduction of the deformations and von Mises stresses (48.21%) at the fixator tree (Table 7).
Figure 18. Comparison diagram of displacements at the place of fracture R and at the place of load for initial and optimized design. Significant decreases of displacement of bone segment Dp (68.31%) (Figure 16) and interfragmentary displacement Dg (R) (Figure 18) are achieved by increasing the diameter of the external fixation device tree (56.37%), which results in a significant increase of the resistance moment (108.52%), and a reduction of the deformations and von Mises stresses (48.21%) at the fixator tree (Table 7).
Figure 19. Initial (a) and optimized (b) design of external fixation device. The results of the KBE system for size optimization on the real design of externa fixation device are shown on Figures 14—20 and Tables 7 and 8.
Figure 19. Initial (a) and optimized (b) design of external fixation device. The results of the KBE system for size optimization on the real design of externa fixation device are shown on Figures 14—20 and Tables 7 and 8.
Figure 20. Distribution of von Mises stresses (a) and principal stresses (b) on initial and optimized design of the external fixation device.
Figure 20. Distribution of von Mises stresses (a) and principal stresses (b) on initial and optimized design of the external fixation device.
Table 8. Distribution of von Mises stresses on fixator components for initial and optimized design.
Table 8. Distribution of von Mises stresses on fixator components for initial and optimized design.
Coupling body with maximal von Mises stress (position 1 on Figure 20a) Coupling carrier body with maximal von Mises stress (position 2 on Figure 20a)
Coupling body with maximal von Mises stress (position 1 on Figure 20a) Coupling carrier body with maximal von Mises stress (position 2 on Figure 20a)
Related topics:
Experimental Stress AnalysisKnowledge Base EngineeringExternal FixationSize and Shape Optimization
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