Optimized punch contact action related to control of local structure displacement

For a structure under service loads, there is a need to induce precise control of a local displacement by additional punch loading. Such problem exists in design of robot grippers or agricultural tools used in mechanical processing. The punch interaction is assumed to be executed by a discrete set of pins or by a continuously distributed contact pressure. The optimal contact force or pressure distribution and contact shape are specified for both discrete and continuous punch action. Several boundary support conditions are discussed, and their effects on punch action are presented.


Introduction
The structural optimization problems are usually related to specification of material parameters, size, shape, or topological variables, loading distribution, supports, element connections, etc. The required stiffness and strength levels are usually imposed as design constraints and the objective function assumed as the volume or cost of materials used, cf. recent books by Banichuk and Neittaanmaki (2010) or Banichuk (2011). A special class of contact optimization problems is characterized by searching for optimal contact traction distribution satisfying strength conditions at the contact interface. The review of contact optimization problems has been presented by Páczelt et al. (2016).
In the paper by Li et al. (2003), the evolutionary structural optimization (ESO) concepts have been applied with a nongradient procedure presented for incremental shape redesign of contact interfaces. In Zabaras et al.'s (2000) work, a continuum sensitivity analysis has been presented for large inelastic deformations and metal-forming processes.
In the papers by Páczelt and Szabó (1994), Páczelt (2000), Páczelt and Baksa (2002), and Páczelt et al. (2007), the method of contact shape optimization was developed for 2D and 3D problems with the objective to minimize the maximal contact pressure under specified loading conditions. The contact optimization problems for elements in the relative sliding motion with account for wear require special analysis referred to a steady-state wear process. The optimal design objective is to specify the contact shape minimizing the wear rate in the steady-state condition for progressive or periodic sliding. The extensive review of this class of problems has been presented by Páczelt et al. (2015). The monographs of Goryacheva (1998) and Wriggers (2002) provide a solid foundation for analytical and numerical methods of solution of contact problems, including wear analysis. The finite element analysis is most frequently applied in solving contact problems, cf. Szabó and Babuska (1991).
The present paper is devoted to a problem of local displacement control in a structure subjected to service loads, such as an assembling robot gripper (Monkman et al. 2006). Then the precise displacement value should be achieved at the location of robot interaction with an assembled element. It means that the displacement control should be applied at a point transmitting the interaction load. The method of solution of this new class of contact design problems will be discussed in Sect. 2 for the case of beam deflection control with the constraint set on maximal contact pressure. In Sect. 3, the specific examples are discussed, illustrating designs of punch shape for differing beam support constraints. In Sect. 4, the optimal contact Responsible Editor: Somanath Nagendra pressure distribution and punch shape are determined from the optimization procedure. Applying the strength constraint also punch position and contact zone size can be properly selected. The method presented can be extended to the analysis and design of optimal punch interaction aimed at displacement control at the loaded boundary point of any structural element.
2 Control of beam deflection at the loading point Q for differing support conditions In the assembling process of mechanical elements conducted by a robot gripper, a typical operation is to place one body (cylinder) into a hole of another body. In this case, the cylinder must execute translation u * n under increasing axial force. In Fig. 1a, the force F Q is equal to friction force at the interface between cylinder and hole. Another operation called "pick and place" is related to lifting the cylinder and placing in a new position, Fig. 1b. In this case, the cylinder is compressed along its diameter by two plates inducing normal contact displacements u * n non-linearly related to contact forces F Q , assuring due to friction a fixed cylinder position relative to plates. In both operations, the prescribed force and displacement values are required at the same point.
Consider a cantilever beam built-in at its end A and loaded at point Q by the force F Q , Fig. 2, inducing the deflection u 2 ð Þ n . To keep the deflection of Q at the required value u * n , the discrete or continuous punch action is applied within the specified contact zone region Ω defined as an interval L 1 ≤ x ≤ L 4 and the contact center position x = x 0 . The punch is allowed to translate in the normal direction n to the beam exerting contact pressure p n (x). The desired punch action occurs when the value of punch load F 0 is reached for the contact pressure distribution not exceeding the specified level p max . The value of F 0 depends on the size and position of the contact zone. The minimal value of F 0 is reached, when the concentrated load acts at x = L 4 , but when the constraint p n ≤ p max is applied, then the constant pressure zone p n = p max at the boundary x = L 4 constitutes the optimal distribution. In our design, the location of the punch and the control function c(x) are prescribed in advance and the value of p max is controlled by the length of contact zone. The prescribed pressure distribution p n = c(x) p max is reached by determination of the initial gap between the punch and beam. The problem is solved in two main steps. First, using the stamp equilibrium condition at a given control function c(x) and constraint of normal displacement at the point Q, the value of p max is specified, see (6). Second, from contact condition between the stamp and beam, the initial gap between the contacting bodies is determined, see Appendix 1.
The constraint set on the maximal pressure value can be attained by assuming the following distribution in the contact zone specified by the control function Further, for technical reasons, it is assumed that the contact pressure distribution is symmetric with respect to center point of contact zone. This allows for punch application in differing support conditions. Then obviously, the optimal distribution would correspond to c(x) = 1 and constant pressure p n = p max ; next, this value could be minimized. Figure 3 presents the assumed symmetric pressure distribution for different control functions. Using Hermite functions, two pressure distributions are specified. The formulae of Hermite functions are stated in the caption of Fig. 3.
Assuming the pressure distributions according to Fig. 3, the problem is reduced to specification of contact surface form  represented by the gap function. Several variants of this problem are treated and will be discussed in more detail.
The punch center position x 0 can be located on both sides of point Q. Depending on the support constraint, it can essentially affect the value of the required punch load F 0 or the deformation form. The effects of punch position will be discussed in Sect. 4.
for Hermite 2: L 1 = 300, L 2 = L 3 = 450, L 4 = 600 2.1 Beam structure deflection control for different support conditions 2.1.1 First variant: cantilever beam built-in at the end a Consider first the case of punch action executed by a set of punch pins, as shown in Fig. 2a, of thickness a, width b, and cross-section area A = ab. The forces between punch and beam are specified by (1), thus Under applied load F 0 on the punch, the maximal pressure results from the equilibrium condition H i ð Þ x; s ð Þp n s ð Þds is the normal displacement of beam or punch induced by the contact pressure, H (i) (x, s) is the Green function for normal displacements, u 2 ð Þ n;load is the normal displacement resulted from the load F Q , g (0) is the initial gap, λ is the rigid body displacement of the punch, that is d = d(p n , λ), where the notation λ Usually, the contact problem is solved by the principle of modified complementary energy or by the displacement formulation using minimum principle of the total potential energy in the penalty form (cf. Wriggers 2002), (Konyukhov and Schweizerhof (2013). In the last case, we use p-versions of finite element method for discretization (cf. Szabó and Babuska 1991).
In these contact optimization problems, the initial gap (shape form of the contact surface) is the unknown function. In our case, the contact condition will be checked up at the kont points. Supposing contact at each check point, the calculation of the gap can be executed in terms of the special iterative procedure (Páczelt 2000;Páczelt et al. 2016) In Appendices 1 and 2, the discretized equations for determination of the initial gap between the beam and punch are provided.
2.1.2 Second variant: beam allowed to execute displacement along the sliding support A Let us now refer to the system of Fig. 4, where beam 2 can execute the rigid body displacement −u A;z ¼ λ 2 ð Þ F along the sliding support A and deform elastically due to load F Q and punch force F 0 action. Assume the initial beam configuration to correspond to contact engagement at Q with an object (e.g., cylinder).The vertical displacement at the point Q then is Assuming that there is no force acting at the sliding support A, the equilibrium condition of beam 2 requires that as the contact forces are P j = Ac(x j )p max .
Using the influence function for calculation of u 2 ð Þ n;load , we can write and the support displacement can be determined from Eq.(8) The contact condition between the punch and beam is Using (62a), (12) can be expressed as follows where Discretizing (12)-(15), we can write  (11)), then iter u can be calculated, namely and from the following equation one can easily find iter λ 1 ð Þ F ; ðiterÞ g 0 ð Þ supposing (iter) g 1 (0) = 0.

Third variant: free beam is allowed for rigid body translation and rotation
Referring to Fig. 5, consider the case, when the beam 2 is not constrained and is allowed for a rigid body vertical displacement λ 2 ð Þ F and rotation λ 2 ð Þ M at the beam end A. The displacement control is now required at two points Q 1 and Q 2 , where the loads F Q 1 and F Q 2 are applied. The initial beam configuration corresponds to simultaneous contact engagement with interacting objects at Q 1 and Q 2 .
The vertical displacement at point Q s is Using the influence functions for calculation u 2 ð Þ sn;load , we can write Beam and punch system for transmission of load F 0 interacting with the force F Q to induce vertical displacement u * n . Beam 2 is allowed for rigid body displacement λ Beam system for transmission of the load F 0 interacting with the forces F Q 1 and F Q 2 to induce vertical displacements u * 1n and u * 2n , when the beam 2 is allowed for rigid body translation λ 2 ð Þ F and rotation λ 2 ð Þ M From the force and moment equilibrium equations of the beam, we have and the position of control force F 0 is equal to Introducing the parameter 0 ≤ ζ ≤ 1 and defining the force when ζ = 0, the forces From the equilibrium equation for punch (3), the maximal contact pressure and the contact forces are For derivation of the rigid body displacements, we have the following two equations The contact condition between the punch and beam is and it can be expressed in the following form Discretizing (26)-(28), we can write where.
Because F 0 and F Q p p = 1, 2 and λ 2 ð Þ F and λ 2 ð Þ M are known, iter u can be calculated from the equation and from the following equation one can easily find iter λ

Numerical examples
The beam area A b = a b h b = 20 ⋅ 50 = 1000 mm 2 , inertia moment I ¼ a b h 3 b =12 ¼ 21833:33 mm 4 , Young modulus E = 2 ⋅ 10 5 MPa are assumed and other geometric parameters are: L 1 = 300, L 4 = 600, x Q = 850, L = 900 mm. The punch centre position x 0 = 450 mm corresponds to the position parameter ξ = x 0 /x Q = 0.529. The specified vertical displacement at the point Q is u * n ¼ 1 mm, and the required force values are F Q = 4 kN, F Q = 5 kN, and F Q = 6 kN.
The beam deflection is presented in Fig. 6a for three control functions and F Q = 5 kN. It is seen that their effect is very small. The biggest displacement corresponds to the Hermite 2 control function (see Fig. 3). At the point Q, the prescribed displacement u * n ¼ 1 mm at x = 850 mm is attained. Figure 6b presents the beam deflection for three values of force F Q . The heights of punch pins at the start of calculation are assumed to be 50 mm. From the numerical solution the new heights are calculated by (68) and depicted in Fig. 7a.
The initial gaps at discrete points in Fig. 7b illustrate the effect of load values on punch profile at constant contact pressure. It is seen that the height differences become more significant at larger distance from the force F Q application. The maximal height difference is less than 0.15 mm.
The distribution of contact tractions is presented in Fig. 8 for F Q = 5 kN. The maximum value is reached for the control function Hermite 2, the least value for c(x) = const = 1. In the latter case the contact force is the same at each pin: P j = 2.987 kN and the punch load equals F 0 = 14.935 kN. This means that the present design requires the considerably higher load to execute the control at the efficiency load factor f 0 ¼ F 0 F Q ¼ 2:987. At other force levels F Q = 4 kN, F Q = 6 kN, the maximal contact forces (c(x) = const = 1) are P jmax = 2.41 kN, P jmax = 3.56 kN, respectively, and the external punch loads in these cases are F 0 = 12.07 kN, F 0 = 17.81 kN, corresponding to load factors f 0 = 3.0 and 2.987.

Model 2: continuous contact interaction
Consider now the continuous contact interaction for the elastic punch in the plane stress state. The punch interaction is performed in the segment 420 ≤ x ≤ 480 (L 1 = 420, L 4 = 480), and its height is 50 mm (see Figure 2b). The uniform pressure distribution, c(x) = 1, was assumed in the numerical solution and the normal displacement u 1 ð Þ n x; p n ð Þ was calculated by the finite element method, using pversion technique, cf. Szabó and Babuska (1991). The punch is loaded by pressure p n = p max at the contact boundary z = 0 and by the same pressure at the upper boundary for z = 50 mm. The elastic boundary value problem has been solved for differing finite element meshes (n x = 4, 8, 12; n z = 5, where n x , n z is number of elements in x and z directions) using polynomial order 3 ≤ p ≤ 8 for quadrilateral finite elements with the shape functions in the trunk space. NDOF at n x = 4, n z = 5, p = 3, 8 is equal to 135, 735 at n x = 8, n z = 5, p = 3, 8 is 270, 1470. The normal displacement distribution on the boundary surfaces exhibits good convergence for increasing p. For p ≥ 6, the numerical results differ very little for all meshes, their difference being smaller than 0.15%.
The contact conditions are checked at the Lobatto integral points. The discretized problem is solved in the following steps: 1. Calculation of the contact pressure by (6), and F 0 calculated by (3).
Þ ) i n p u n c h from a separate solution (see (70)). 4. From Signorini contact condition: d = 0, we can find initial gap iter g (0) and rigid body displacement iter λ (see (72), (65c), (66). 5. As the initial gap modifies punch shape, the iteration process must be used. The steps 3 and 4 are continued until The rigid body vertical displacement of punch, λ ¼ λ 1 ð Þ F , is a displacement in the -z direction at point x = 420, z = 50. In our case for F Q = 5 kN, the final value is λ 1 ð Þ F ¼ 2:511 mm. The deflection and initial gap are presented in Fig. 9a, b for all values of forces F Q . For the optimal solution, the values the load F 0 and pressure p max are collected in Table 1.
The beam deflection profiles for both discrete and continuous punch actions are very similar (see Figs. 6 and 9). Note that the contact zone for discrete punch action is much larger (300 mm), than that for continuous action (60 mm) at the same centre position x = 450 mm. The force efficiency factors in all cases are practically equal to 3.0.

Example for the second variant: allowed beam support translation
In this case, the load equilibrium requires that F 0 = F Q (cf. Figure 4). In the first step, the rigid body displacement λ 2 ð Þ F is calculated from Eq. (11). From contact condition between beam and punch, the rigid body displacement λ 1 ð Þ F is determined from (18). Only several iterations are required to reach the solution. The beam deflection is shown in Fig. 10a. The punch pins profile is presented in Fig. 10b.
The force efficiency factor f 0 = F 0 /F Q in this case is equal to 1. The rigid body punch and beam displacements are collected in Table 2 at different F Q .

Example for 3rd variant: beam allowed for rigid body translation and rotation
Consider the beam of crosssection dimensions a b = 20, h b = 75 mm (cf. Fig. 5). In this case, the load on the punch is F Q 1 ¼ 4; 5; 6 kN , and ζ = 0.5. The prescribed vertical displacements at the points Q 1 and Q 2 are assumed: u • 1n ¼ 5 mm and u • 2n ¼ 10 mm. Let us note that in the case of contact interaction at Q 1 and Q 2 , the displacements are specified from the loadpenetration rules. The punch center coordinate is.
In the first step, the rigid body motion components λ    Fig. 11b. The punch vertical translation λ 1 ð Þ F and the loading force F 0 are therefore required for proper operation of gripper at the points Q 1 and Q 2 . The rigid body displacements are collected in Table 3. Let us note that for varying control requirements at the contact points, the sliding or rotating punch can be applied assuring proper load F 0 position at each case.

Localized and distributed punch action: effect of design constraints
The numerical examples presented in Sect. 3 illustrate the discrete and continuous punch action in order to control the deflection at point Q (or at several points, as discussed in variant 3). In the design of such control system, the punch position x 0 and length L 1 − 4 = L 4 − L 1 of the contact zone are important variables, as the applied load value and the maximal contact pressure should be minimized. Also, the maximal beam stress σ max in the loaded state should not exceed the critical stress value, σ max ≤ σ u . Assume first the contact pressure distribution as the design variable subject to optimization.

Contact pressure distribution: optimization for deflection control
Consider the cantilever beam under punch action on the contact zone L 1 − 4 = L 4 − L 1 inducing the deflection w Q = − u z at the location Q. Apply the unit force F = 1 at Q, inducing the state M (q) (x), κ (q) (x), w (q) (x) (bending moment, curvature, deflection) in the beam The independent punch action induces the state M(x), κ(x), and w(x).
The virtual work equation is where M = EIκ, M (q) = EIκ (q) , and M (q) κ = EIκ (q) κ = Mκ (q) , p n is the contact pressure between punch and beam, a b is the beam crosssection width in direction y, h b is the beam crosssection height in direction z, I ¼ a b h 3 b =12 inertia moment of the cross section, and E is the Young's modulus.
From (32), we have Now, several optimization problems can be stated Taking the functional its variation provides the stationary condition It is seen that the problem formulation of optimal pressure distribution has no extremum, since the displacement w (q) is not constant, but λ is assumed constant. There is only the singular optimal solution for the concentrated contact force action located at w Qmax in the contact zone.

Problem 2:
The same result as for problem 1. Problem 3: Assume the penalty approach and the contact pressure functional where p 0 is the reference pressure, p 0 ≈ p max , and p 0 ≤ p max . Then The variation of L p3 has the following form and the stationary conditions are as follows The value of λ * is specified from the equality Consider now the case of symmetric pressure distribution relative to the contact zone center, assuming that where ξ specifies the position of two symmetrically located points.
The virtual work relation now is Denotẽ then we have and from (40), there is The analysis presented characterizes the optimal contact pressure distribution. Figure 12 presents the optimal pressure distribution for varying value of the exponent m and minimization of the functional I p3 . It is seen that for m = 4, the pressure tends to the uniform distribution for both unconstrained and symmetry constrained cases. If there is no constraint set on the maximal pressure value, then the concentrated load action is optimal. Its value and position are determined by requiring the deflection u z ¼ −u * n at Q and accounting for the beam stress constraint σ max ≤ σ u . When the distributed punch action is considered, and the integral constraint is set on the contact pressure, then its distribution is uniform on the whole contact zone. The required punch load for specified deflection u z ¼ −u * n will then be higher from the concentrated load value.

Concentrated and distributed cantilever beam control
Referring to Fig. 2, consider the cantilever beam under the action of concentrated load F Q and the punch action is  Table 3 The maximal contact pressure and rigid body displacements resulting from solution of variant 3 simulated by the concentrated load F 0 . Assume first that the load positions satisfy the inequality x 0 ≤ x Q . The bending moment in the beam is expressed as follows where The normal deflection at the point Q (along-zaxis) due to force F Q (can be expressed from the formulae of Appendix 3) equals The deflection w 0 Q at Q due to load F 0 is Introducing the nondimensional variables Now, we can express the total deflection w t ¼ w Q þ w 0 Q at Q as follows For the load F 0 acting on the right side of Q, x Q ≤ x 0 , the total deflection at Q equals For specifiedw t , the required value of load F 0 is expressed as follows Atw t ¼ 0 and w 0 Q þ w Q ¼ 0, this means that there is no interaction of gripper with the assembled element. For contact interaction, the relative penetration depth equals δ = w t and jw Q j < w 0 Q . The diagrams f 0 (ξ) for positive values of δ = w t > 0 can be plotted. The region above the curve forw t ¼ 0 (see Fig. 13) corresponds to contact interaction. The load efficiency factor f 0 = F 0 /F Q strongly depends on the punch load F 0 position. Usually, the value of ξ = x 0 /x Q is selected in a specific design case with account for contact pressure and beam strength constraints. For instance, assume the punch load to belong to segments 0.4 ≤ ξ ≤ 0.7 or 1.2 ≤ ξ ≤ 1.5. Then the point ξ = 450/850 = 0.529, f 0 ≈ 3, marked in Fig. 13 (point B) represents the design discussed in Sect. 3.1, cf. (Fig. 9), for F Q = 5 kN, w t ¼ u * n ¼ 1 mm, jw Q j ¼ ju 2 ð Þ n;load j ¼ 24:56 mm, andw t ≈0:04. Consider the strength constraint. The critical bending moment results from the stress constraint M ≤ M c ¼ a b h 2 b σ u =6, where σ u is the ultimate stress. DenotingM ¼ M = F Q x Q À Á , the maximal bending moment for ξ ≤ 1 is reached at the support x = 0 and its absolute value equals For ξ ≥ 1, the maximal bending moment is reached at Q and its absolute value is Satisfying the strength conditions jM b j ≤M c and jM Q j≤M c , in view of (52), (53), and (54), the limits of applicable load positions are expressed by the inequalities Figure 13 shows the admissible segment of punch action satisfying the constraints (55) ð Þ =x Q affects the resultant punch load relative to the concentrated load action at the same position. The required punch load value is higher for the same value of induced displacementw t .

Translating beam control under concentrated loads, variant 2
Consider the beam of Fig. 4 with the sliding support at the left end, loaded by the concentrated load F 0 = F Q at the distance x 0 and the specified contact force F Q at Q located at the distance x Q .
Consider first the case x 0 ≤ x Q or ξ ≤ 1. At the support, the transverse force equals F b = 0 and the moment is The deflection form now is expressed as follows For where w b ¼ λ 2 ð Þ F denotes the beam translation at the support. Requiring w = w t at Q, the beam translation w b is determined, thus wherẽ The Þcan easily be constructed, see Fig. 14. It is seen that the sliding displacement changes sign at ξ = 1.
The present sliding support case differs essentially from the previous builtin case. The applied punch force is constant and does not depend on the position, but the deflection form radically changes when passing from ξ ≤ 1 to ξ ≥ 1. Figure 15 presents both deflection forms for concentrated (Fig. 15a,  and distributed (Fig. 15c) punch action. The present case is convenient for more precise control at Q by requiring both deflection and slope to be kept at specified values and allowing for sliding punch action. In Fig. 15a, c the solutions of the optimization problems satisfying stress constraint are presented. In Fig. 15 the optimal solution point is marked by o.

Concluding remarks
The optimal design of punch action in order to control normal displacement at a loaded boundary point in a structural element has been discussed in the paper and illustrated by the specific examples of beam deflection control. It was demonstrated that the support constraint can affect essentially the punch load and the beam deflected form. This new class of problems is characterized by a set of design variables, such as contact pressure distribution p n (x), punch resultant load F 0 , punch centre position x 0 , and the size of contact zone L 1 − 4 . The minimization of punch load or the maximal contact pressure, required for displacement control, has been discussed, also the effect of punch position and contact zone size on the punch load value was considered.
The present problem formulation can be extended to more advanced control problems. First, for the gripper operation requiring varying load and displacement control at Q, or following the loading path F 0 ¼ F 0 u * n À Á , the punch action should be executed for properly varying load F 0 . The other extension is related to control of both the deflection and its slope at Q. Such control can be performed by varying loads of two punches. Applying such punch action to plate elements, the normal displacement and the orientation angles of the normal vector to the deflected surface at Q can be controlled. Any tool attached transversely at Q to the plate could then execute both normal and inplane displacements. Such advanced control problems will be discussed in a separate paper.

Replication of results
The computer codes are written in FORTRAN and MATLAB in the research form. To use them without additional comments could be complicated. If anybody is interested in the programs, please write to István Páczelt.
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