Subject of Investigation
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- Abigail Spencer
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1 Subject of Investigation A 3D Analysis of the Node to Node and Node to Surface Contact Phenomenon using Penalty Function and Lagrange Multipliers Constraints in Matlab Academic Advisor Professor José M. A. César de Sá Authors José M. V. Berardo Behzad V. Farahani Master s Students of Computational Mechanics Department of Mechanical Engineering Faculty of Engineering University of Porto January 2015
2 Contents List of figures and tables... 2 Abstract Introduction to Contact Theory Node to Node Contact Node to Surface Contact Surface to Surface Contact Node to Segment Contact Segment to Segment Contact Treatment of Contact Constraints; Mathematics Lagrange Multipliers Penalty Function Augmented Lagrangian method Mechanical Examples relevant to Contact First initial example, Node to Node contact method Node to node contact without any methods Node to node contact with the Lagrange multipliers constraint Node to node contact with the Penalty function constraint Second initial example, Node to Surface contact method Node to surface contact without any methods Node to surface contact of the Penalty function constraint Further examples of the contact methods First Benchmark - Node to Node Contact, Change in Mesh Second Benchmark - Node to Node Contact, Applying Angle Third Benchmark - Node to surface Contact, Change in Mesh Conclusion and discussion Final Remarks Reference JANUARY 2015 BY JOSÉ BERARDO & BEHZAD FARAHANI 1
3 List of figures and tables Figure 1: Depiction of contact detection for contact/impact problems... 4 Figure 2: Node to node contact status... 5 Figure 3: schematic view of Node to surface contact... 6 Figure 4: Proper mesh for contact analysis in (a) Linear stress and (b) nonlinear stress Analysis... 7 Figure 5: Node to segment contact element geometry... 8 Figure 6: Schematic view of segment to segment contact... 8 Figure 7: Schematic view of the geometry of problem studied in the investigation...11 Figure 8: Graphical representation of the first initial example (a) Labeling the number of elements and (b) Labeling the number of nodes...14 Figure 9: Deformed mesh of the solid prisms for the first example with no contact methods...14 Figure 10: The corresponding faces involved in contact procedure for node to node manner...15 Figure 11: Deformed shape of the prisms for node to node contact in Lagrange Multipliers method...17 Figure 12: Deformed shape of the prisms for node to node contact in Penalty function method...19 Figure 13: Geometry of the second example node to surface state for (a) Number of total elements and (b) number of total nodes...20 Figure 14: Deformed mesh of the second example when no constraints applied...21 Figure 15: Contact faces in both prisms with the nodes located at in local coordinate system...22 Figure 16 : Deformed mesh of the second example node to surface with Penalty function method...24 Figure 17: The position of node out of the face and its projection on the face to obtain the shortest distance.25 Figure 18: The geometry created for first benchmark, (a) Node labeling and (b) Element labeling...27 Figure 19: Deformed shape of geometry of the first benchmark associated with node to node contact...28 Figure 20: The stress field on the prisms for the first benchmark on (a) Solid body and (b) Cross sections.28 Figure 21 : The geometry created for second benchmark, (a) Node labeling and (b) Element labeling...30 Figure 22 : Deformed shape of geometry of the second benchmark associated with node to node contact..30 Figure 23: Equivalent von Mises stress on the super convergent points for the second benchmark...31 Figure 24 : The geometry created for third benchmark, (a) Node labeling and (b) Element labeling...32 Figure 25: Deformed shape of geometry of the third benchmark associated with node to surface contact...33 Figure 26: The displacement field on prisms for the third benchmark on (a) Solid body and (b) Cross sections...34 Figure 27: Deformed shape of geometry of the third benchmark associated with angle = 5 degrees...35 Figure 28: Deformed shape of the prisms for node to node contact in the discussion example...36 Table 1: Material properties related to the problem Table 2: Geometrical values related to the first initial example Table 3: The geometrical characteristics related to second example Table 4: The characteristics of the first benchmark, node to node contact Table 5: The characteristics of the second benchmark node to node contact Table 6: The characteristics of the third benchmark node to surface contact Table 7: The characteristics of the discussion example node to node contact JANUARY 2015 BY JOSÉ BERARDO & BEHZAD FARAHANI 2
4 Abstract Since this study is done in the frame work of the contact mechanic task, its theoretical part is explained briefly first. It contains some information regarding the relevant different contact methods and constraint techniques. Then, it is going to present some mechanical examples which are programed based on the finite element method formulations in Matlab. Based on the purpose of the project, two different constraint methods are applied to solve the problem; Penalty Function and Lagrange multipliers. Then, two different methods of contact are considered for this study; Node to Node and Node to Surface. In the perspective of the code, it is possible to generate automatic code of Node to Surface (any node or length combinations between solid bodies) for penalty method only. In the case of Node to Node, the mentioned aspect is possible for both Penalty and Lagrange multipliers manners. JANUARY 2015 BY JOSÉ BERARDO & BEHZAD FARAHANI 3
5 1- Introduction to Contact Theory The current study is relevant to the contact mechanics in computational mechanics course. Thus, it is focused on the computational concepts of the problem. Basically, the contact problems are relevant to the contact phenomenon between rigid and deformable bodies, between deformable bodies or even self-contact in a deformable body. Contact detection is also known as contact searching. The collection of nodes that are in contact are called the active set. The purpose of detection is to determine the contact area so that contact constraints are correctly applied. For quasi-static problems, the detection process is done by monitoring the change of gaps g. When the gap g of a contact pair changes from negative to zero or positive, this pair is regarded as being in contact and added into the active set. Figure 1: Depiction of contact detection for contact/impact problems Figure 1 shows the strategy of contact detection for impact problems. Note the gap shown here is artificially enlarged for clear illustration. Currently, there are two important aspects should be paid attention to as follows: Non-penetration; Friction. The first important issue is regarding the non-penetration phenomenon. It means that some constrain should be defined in the problem to achieve the non-penetration state in the contact procedure. Second aspect, it is possible to apply the friction to the concept of the problem to study the contact problems. Based on the purpose of the study, non-penetration JANUARY 2015 BY JOSÉ BERARDO & BEHZAD FARAHANI 4
6 status is desirable. So, it is needed to present some explanations concerning its definition and mathematical formulations. There exist several types of contact method as follows; Node to node; Node to surface; Surface to surface; Node to segment; Segment to segment Node to Node Contact Basically, it is available only for faces that are initially touching and it is needed to create coincident nodes on source and target faces in a compatible mesh. This contact formulation is available with the No Penetration contact type for static, nonlinear, and thermal studies. For static and nonlinear studies, this contact type prevents interference between the source and target faces but allows them to move away from each other to form gaps. Generally, this formulation might be faster than node to surface and surface to surface approaches, but it is the least accurate for general structural problems with sliding or large rotations. The accuracy of the results depends on the loading, being best if the two faces are pressed against each other without much sliding or relative rotations. The accuracy reduces as the loading causes large sliding or rotations. Figure 2: Node to node contact status JANUARY 2015 BY JOSÉ BERARDO & BEHZAD FARAHANI 5
7 Figure 2 illustrates a possible deformation of two faces that were initially touching Node to Surface Contact Basically, node to surface contact elements are used to model contact between a surface and a point for example a sharp object like a pin or bullet impacting a plate, membrane and etc. Indeed, it does not require that the faces are initially touching. Moreover, there is no need to create compatible mesh between source and target faces. This contact formulation is available with the No Penetration contact type for static, nonlinear, and thermal studies. For static and nonlinear studies, this contact type prevents interference between the source and target faces but allows them to move away from each other to form gaps. Figure 3: schematic view of Node to surface contact Figure 3 demonstrates a possible deformation of two faces with incompatible mesh. It prevents interference but allows separation. JANUARY 2015 BY JOSÉ BERARDO & BEHZAD FARAHANI 6
8 1-3- Surface to Surface Contact In fact, surface to surface contact elements are used to model contact between two specific surfaces. As an illustration, two blocks moving relative to each other, contact between two concentric cylinders, contact between car and road and etc. Actually, it is more general than node to node and node to surface contacts. Suitable for complex contacts with general loading. It does require compatible mesh between source and target faces. In addition, it allows faces only as source and target entities. Figure 4: Proper mesh for contact analysis in (a) Linear stress and (b) nonlinear stress Analysis Although surface to surface contact is more accurate in general, the node to surface option gives better results if the contact area between the two faces becomes very small or reduce to a line or point Node to Segment Contact The so-called node to segment contact is probably the most widely used discretization technique for large deformation contact between surfaces with non-matching meshes. The main issues encountered with the node to segment contact formulation are caused by the relative tangential displacement of the contact surfaces. The measure of the penetration is calculated by the closest point projection from the slave node onto the master surface. The normal force is then defined to be coaxial with the normal of the master face at the corresponding location. JANUARY 2015 BY JOSÉ BERARDO & BEHZAD FARAHANI 7
9 Figure 5: Node to segment contact element geometry The node to segment approach enforces contact using collocation such that nodes on the slave side must not penetrate their opposing master side segments. It is frequently applied on a line or surface segment while it allows unequal number of elements at both sides of contact surface. Moreover, it could be useful in the problems with normal definition near corners and for the nodes sliding to a next segment Segment to Segment Contact In the segment to segment contact definition, the contact elements are created between the slave nodes within one face and the nodes of the opposite master face. Generally, each contact segment has a number of auxiliary points, which are located at fixed positions on the contact segment and which are only used during the contact detection phase (figure 6). Figure 6: Schematic view of segment to segment contact It might be useful in the beam to beam contact problems and if the large amount of sliding exists. It is also applicable for specify normal direction while the cross sections can be JANUARY 2015 BY JOSÉ BERARDO & BEHZAD FARAHANI 8
10 accounted for in closure distance determination. Particularly, they are used to detect contact between potential contact segments. According to figure 6, it is noticeable that the displacement is the same along the nodes # 1 and # 2. Moreover, penetration may be limited by a correction procedure through applying Augmented Lagrangian method which is presented in the next section. 2- Treatment of Contact Constraints; Mathematics Contact problems are usually treated as constrained energy minimization problems and optimization techniques are applicable to reach the solution. Despite various approaches proposed in the literature, the following four approaches are well established and are the most widely accepted. Lagrange multipliers; Penalty function; The perturbed Lagrangian method; Augmented Lagrange method. The concepts behind these methods are briefly introduced in this section. Frictionless condition and linear deformation are assumed in the description Lagrange Multipliers The total energy Π total in a two-body contact problem contains two parts wherein the first part (Π 1 + Π 2 ) comprises the kinetic energy and strain energy, and the second part comes from contribution of the contact tractions. To wit, Π total = Π 1 + Π 2 + λ N gda (1) Γ c where; JANUARY 2015 BY JOSÉ BERARDO & BEHZAD FARAHANI 9
11 λ N is the Lagrange multiplier. Theoretically, λ N coincides with the normal contact traction t N in the Lagrange multiplier approach. If the Kuhn-Tucker condition is satisfied exactly, λ N gda Γ c adds nothing to the total energy. The main advantage of the Lagrange multiplier method is that the impenetrability is satisfied almost perfectly. The main drawback of this approach is the possibility of loss of positive definiteness due to the zero diagonal parts in the discretized governing equations. Determination of sticking or sliding status also may present difficulties when friction is present Penalty Function Vanishing of the gap function g = 0 represents the ideal condition that no penetration is allowed, and a negative gap g < 0 represents the allowable configurations in release. In the penalty approach, the inequality of the gap function is relaxed and g can take positive values but the energy function is penalized when penetration occurs. To wit, the original variational inequality becomes an unconstrained extremum. The penalized energy formulation of the two-body contact problems reads as: where Π total = Π 1 + Π ε 2 N g 2 da (2) Γ c ε N > 0 is the penalty parameter in the normal direction. The symbol is the Macauley bracket with the property x if x 0 x = { x if x < Augmented Lagrangian method (3) The Augmented Lagrangian approach combines the concepts of the penalty and the Lagrange multipliers methods. It is an iterative procedure which the Lagrange multiplier is kept constant in each iteration and is only updated at the end of each step. JANUARY 2015 BY JOSÉ BERARDO & BEHZAD FARAHANI 10
12 Π total = Π 1 + Π 2 + [ 1 λ 2ε N + ε N g 2 1 λ 2 N 2ε N ] da (4) Γ c N The advantage of this method is that the penalty parameter may be smaller than in the case of the penalty method, thus avoiding ill-conditioning problems. 3- Mechanical Examples relevant to Contact In this section, it is focused on two prisms which have a contact with each other. Figure 7 shows the schematic view of the geometry of the problem studied in this project. As it is shown, there are two prisms with specific dimensions where the force applied at z direction. Figure 7: Schematic view of the geometry of problem studied in the investigation Based on the above-mentioned geometry an in accordance with the contact formulas, the problem is programed in Matlab so as to study the contact phenomenon. The code possesses some special features as follows: 1- It is possible to increase the number of nodes and elements on each prism and get the results. 2- Based on the logic of the code, it can create the automatic mesh and the coordinate generator in the problem. Then, the program attempts to solve the problem based on the mathematical relationships defined according to the corresponding approach that it is choosable. 3- It has the potential to generate automatic code of Node-to-surface (any node or length combinations between solid bodies) for penalty method only. This foregoing JANUARY 2015 BY JOSÉ BERARDO & BEHZAD FARAHANI 11
13 demanded an algorithm to check if a node from the down body top surface as a point of a bottom face of an element from the bottom face of the upper body. 4- It is also possible to generate automatically from Node-to-node only in the total number of nodes sense, but it s needed that each face of contact faces has the same number of nodes, both methods are available, knowing the penalty method and the Lagrange multipliers method. Additionally, the angle of slop of top face from the bottom prism could be predefined. It is applied on the top face slope α [0, 15] relevant to the bottom prism. It is demanded an additional algorithm to create the nodes in a slop way and demanded the use of an initial automatic gape algorithm finder to apply between the nodes-to-nodes or from nodes to surface. The program has the potential to convert this angle from degrees to radian format. Apparently, this change in the angle is applied on the all nodes located at the top surface. Moreover, this feature is automatically generated through putting the amount of the angle. According to finite element theory, the element type used in this study is 2-D Bilinear Quadrilateral Elements for each surface of the prisms. Apparently, each node possess 3DOF of the displacement being components (u, v, w) in the direction (x, y, z). Here, some numerical examples are presented, the results are obtained from the code First initial example, Node to Node contact method Logically, there should be a gap between two prisms named as g defined as follows: g = 0.1 min ([L z L 1 z 2 ]) (5) The above relationship represents the total gap. It is possible to obtain the gap associated with each prism, g 1 = g 2 and g 2 = g 2 Where they represent the practical gap relevant to each prism. Apparently, this kind of gap is defined between two faces of the contact zone. Later, it is attempted to define different value of gap related to each node. (6) JANUARY 2015 BY JOSÉ BERARDO & BEHZAD FARAHANI 12
14 Table 1: Material properties related to the problem Parameter Bottom prism #1 Top Prism #2 Young`s modulus E 1 = 1500 [Pa] E 2 = 1500 [Pa] Poison`s ratio ν 1 = 0.3 ν 2 = 0.3 In order to start the procedure of first example, it is better to define the dimensions relevant to two prisms presented on table 2. Table 2: Geometrical values related to the first initial example Parameter Bottom prism #1 Top prism #2 Dimension L x = L y = L z = 1 [m] L x = L y = L z = 1 [m] Angle of top face slope α 1 = 0 Not Defined Total sum force in one direction chosen in the P 1 = 200 [N] P 2 = 200 [N] top face (z direction) Number of element (N. E) 1 = 1 (N. E) 2 = 1 Number of nodes (N. N) 1 = 8 (N. N) 2 = Node to node contact without any methods As mentioned before, the program has the potential for users to use a specific contact method to solve the problem. Even it is possible to ask the program not to use any kind of contact methods. So, the program is run with no contact methods. The graphical representations are presented in the following figure. JANUARY 2015 BY JOSÉ BERARDO & BEHZAD FARAHANI 13
15 (a) (b) Figure 8: Graphical representation of the first initial example (a) Labeling the number of elements and (b) Labeling the number of nodes Figure 9: Deformed mesh of the solid prisms for the first example with no contact methods As seen in figure 9, the penetration phenomenon occurs when the prisms contact each other. It is related to the gap and the value of the applied force. Hence the constraint should be utilized in order to eliminate the penetration. JANUARY 2015 BY JOSÉ BERARDO & BEHZAD FARAHANI 14
16 Node to node contact with the Lagrange multipliers constraint In order to start this process, two contact faces of the prisms should be considered. In fact, they are involving in the node to node contact process. Figure 10 shows them with the arrangement where the nodes located at. Figure 10: The corresponding faces involved in contact procedure for node to node manner According to the theoretical relation for Lagrange multipliers constraint presented in literature, it is possible to write the corresponding energy equation on the nodes shown in figure 10 as follows: Π L = + λ 1 (g (w 5 w 9 )) + λ 2 (g (w 6 w 10 )) + λ 3 (g (w 7 w 11 )) + λ 4 (g (w 8 w 12 )) (7) Now, the derivatives should be taken in terms of Lagrange Coefficient (λ i ) and the displacement components. It is noticeable that the Lagrange coefficient is not constant here, it is a desirable variable should be derived at the end. Π L λ 1 = 0 g = w 5 w 9 Π L λ 3 = 0 g = w 7 w 11 Π L w 5 = 0 λ 1 Π L w 7 = 0 λ 3 Π L λ 2 = 0 g = w 6 w 10 Π L λ 4 = 0 g = w 8 w 12 Π L w 6 = 0 λ 2 Π L w 8 = 0 λ 4 JANUARY 2015 BY JOSÉ BERARDO & BEHZAD FARAHANI 15
17 Π L w 9 = 0 λ 1 Π L w 11 = 0 λ 3 Π L w 10 = 0 λ 2 Π L w 12 = 0 λ 4 It is simple to represent the initial system of equation based on 16 nodes with 48 DOF of displacement related to the problem as follows: [K] Init {U} Init = {F} Init Now, the size of the system changes due to applying the Lagrange Multipliers constraint equations. It means that there exist other 4 variables λ 1, λ 2, λ 3 and λ 4 should be considered in all matrices. For nodes 5 and 9: [K] Lagr. Lagr. Lagr {U} 52 1 = {F} 52 1 K(49,15) = 1 K(49,27) = 1 K(15,49) = 1 K(27,49) = 1 For nodes 6 and 10: K(50,18) = 1 K(50,30) = 1 K(18,50) = 1 K(30,50) = 1 For nodes 7 and 11: K(51,21) = 1 K(51,33) = 1 K(21,51) = 1 K(33,51) = 1 For nodes 8 and 12: K(52,24) = 1 K(52,36) = 1 K(25,52) = 1 K(36,52) = 1 The obtained stiffness matrix is symmetry. Now, it turns to create the corresponding displacement and force vectors. U(49,1) = λ 1 U(50,1) = λ 2 U(51,1) = λ 3 U(52,1) = λ 4 F(49,1) = g F(50,1) = g F(51,1) = g F(52,1) = g where g is the vertical gap between two contact faces. Based on equation (5) and (6), the total amount of gap is: JANUARY 2015 BY JOSÉ BERARDO & BEHZAD FARAHANI 16
18 g = g total = 0.1 (m) so g 1 = g 2 = 0.05 (m) Apparently, g 1 and g 2 are the gap relevant to prism #1 and #2 respectively. The material properties and the dimensions of the prisms are based on tables 1 and 2. So, the new system of equation is created based on the values introduced above. Then, the program is run, the graphical representation of deformed shape is shown below: Figure 11: Deformed shape of the prisms for node to node contact in Lagrange Multipliers method Figure 11 demonstrates that there is no penetration phenomenon occurs during the contact process and besides the values of Lagrange Coefficients are presented below: Node to node contact with the Penalty function constraint In this part, it is focused on the penalty function for node to node contact method. The material properties and dimensions related to prisms presented on table 1 and table 2 respectively. JANUARY 2015 BY JOSÉ BERARDO & BEHZAD FARAHANI 17
19 As shown in figure 10, if the nodes located at the contact faces are considered, it is possible to rewrite the energy equations. It is noticeable to mention that the constant Penalty Coefficient (ε) is utilized on the equation as a constant variable. [K] Pen {U} Pen = {F} Pen The energy equation associated with penalty function is as follows: Π P = ε(g (w 5 w 9 )) ε(g (w 6 w 10 )) ε(g (w 7 w 11 )) ε(g (w 8 w 12 )) 2 (8) Then, the derivatives in terms of displacement components are taken: Π P w 5 = 0 Π P w 6 = 0 Π P w 7 = 0 Π P w 8 = 0 Π P w 9 = 0 εw 5 εw 9 = εg εw 6 εw 10 = εg εw 7 εw 11 = εg εw 8 εw 12 = εg εw 5 + εw 9 = εg Π P w 10 = 0 εw 6 + εw 10 = εg Π P w 11 = 0 Π P w 12 = 0 εw 7 + εw 11 = εg εw 8 + εw 12 = εg Now, these new values should be substituted in the matrices: K(15,15) = ε K(15,27) = ε K(18,18) = ε K(18,30) = ε K(21,21) = ε K(21,33) = ε K(24,24) = ε K(24,36) = ε K(27,15) = ε K(27,27) = ε K(30,18) = ε K(30,30) = ε K(33,21) = ε K(33,33) = ε K(36,24) = ε K(36,36) = ε JANUARY 2015 BY JOSÉ BERARDO & BEHZAD FARAHANI 18
20 The new values relevant to stiffness matrix proves that it is a symmetric matrix. F(15,1) = εg F(18,1) = εg F(21,1) = εg F(24,1) = εg F(27,1) = εg F(30,1) = εg F(33,1) = εg F(36,1) = εg The explanation regarding the gap (g) presented in the last section is true for penalty function method as well. In addition, the initial value of Penalty Coefficient is given as follows: ε initial = 1e + 11 Now, if the program is run, the deformed shape of the prisms is obtained shown in figure 12. Figure 12: Deformed shape of the prisms for node to node contact in Penalty function method Subsequently, the optimal amount of penalty coefficient is updated during the process as follows: ε updated = e + 13 JANUARY 2015 BY JOSÉ BERARDO & BEHZAD FARAHANI 19
21 3-2- Second initial example, Node to Surface contact method In this case, the node to surface contact state is going to be studied based on the theoretical aspect presented in last chapter. In order to start the process, some parameters should be defined as shown in table 3. Indeed, the material properties are the same as table 1. Table 3: The geometrical characteristics related to second example Parameter Bottom prism #1 Top prism #2 Dimension L x = L y = L z = 1 3 [m] L x = L y = L z = 1 [m] Angle of top face slope α 1 = 0 Not Defined Total sum force in one direction chosen in the P 1 = 200 [N] P 2 = 200 [N] top face (z direction) Number of element (N. E) 1 = 1 (N. E) 2 = 1 Number of nodes (N. N) 1 = 8 (N. N) 2 = 8 Now, it is possible to generate the mesh for the problem. Figure 13 shows two prisms related to the current study derived from Matlab. The arrangement of the nodes and the elements are demonstrated in this figure. (a) (b) Figure 13: Geometry of the second example node to surface state for (a) Number of total elements and (b) number of total nodes JANUARY 2015 BY JOSÉ BERARDO & BEHZAD FARAHANI 20
22 Node to surface contact without any methods It is possible to run the program in order to solve the problem without considering any constraints. So, the results related to the deformed mesh are presented as shown in the following figures: Figure 14: Deformed mesh of the second example when no constraints applied In accordance with the above figure, it is clear that the penetration phenomenon occurs during the contact process. It means that the bottom prism went inside the other prism through applying the force on the contact surfaces Node to surface contact of the Penalty function constraint As mentioned in the last sections, there are some specific equations relevant to penalty method should be considered. So, referring to figure 16, if the nodes located at the faces of the contact zone are considered, it is possible to write the energy equation of the penalty method regarding the contact restrains as below: JANUARY 2015 BY JOSÉ BERARDO & BEHZAD FARAHANI 21
23 12 i=9 N i Figure 15: Contact faces in both prisms with the nodes located at in local coordinate system 1 12 ε (g (w 2 5 i=9 N i w i )) 2 (8) w i introduces the displacement of the surface related to the top prism. N i is known as the shape function of the Lagrangian element in local coordinates N 9 = 1 (1 ξ)(1 η) (9-a) 4 N 10 = 1 (1 + ξ)(1 η) (9-b) 4 N 11 = 1 (1 ξ)(1 + η) (9-c) 4 N 12 = 1 (1 + ξ)(1 + η) (9-d) 4 Now, it is needed to calculate the dimension of each nodes on the bottom prism in local coordinate system. For example, if we assume the local dimension for the node 5 as follows: { ξ η } 5 = { } Now, it is possible to obtain the displacement of the similar node of w 5 on the other surface. It is needed to substitute the values of { ξ η } = { 0.5 } in the shape function equations (9) as follows; N 9 = 1 9 ( )( ) = 4 16 N 10 = 1 3 (1 0.5)( ) = 4 16 JANUARY 2015 BY JOSÉ BERARDO & BEHZAD FARAHANI 22
24 N 11 = 1 3 ( )(1 0.5) = 4 16 N 12 = 1 1 (1 0.5)(1 0.5) = 4 16 Apparently, the above relationships are associated with the 5 th node on the face. Now, the energy equation relevant to the penalty method could be written as follows: Π P = + 1 ε (g (w w w w w 16 12)) (10) where ( 9 16 w w w w 12) presents the displacement of the similar node 5 on the top prism. Now, it turns to take the derivatives of the energy equation according to the displacement components: Π P = 0 ε [g (w w 5 ( w w w w 12))] = 0 Π P w 9 = 0 Π P w 10 = 0 Π P w 11 = 0 Π P w 12 = ε [g (w 5 ( 9 16 w w w w 12))] = ε [g (w 5 ( 9 16 w w w w 12))] = ε [g (w 5 ( 9 16 w w w w 12))] = ε [g (w 5 ( 9 16 w w w w 12))] = 0 Now we need to build the stiffness matrix and the force vector based on the values obtained from the derivative equation. [K] {U} 48 1 = {F} 48 1 where [K] represents the global stiffness matrix and {U} and {F} represent the global displacement and force vectors respectively. According to the aforementioned explanation it is possible to substitute the new values in the system of equation related to the problem. JANUARY 2015 BY JOSÉ BERARDO & BEHZAD FARAHANI 23
25 F(15,1) = ε F(27,1) = 9 16 ε F(30,1) = 3 16 ε F(33,1) = 3 16 ε F(36,1) = 1 16 ε K(15,15) = ε K(15,27) = 9 16 ε K(15,30) = 3 16 ε K(15,33) = 3 16 ε K(15,36) = 1 16 ε K(2715) = 9 16 ε K(27,27) = ε K(27,30) = 162 ε K(27,33) = 162 ε K(27,36) = 16 2 ε K(30,15) = ε K(30,27) = ε K(30,30) = 162 ε K(30,33) = 162 ε K(30,36) = 16 2 ε K(33,15) = ε K(33,27) = ε K(33,30) = 162 ε K(33,33) = 162 ε K(33,36) = 16 2 ε K(36,15) = ε K(36,27) = ε K(36,30) = ε K(36,33) = ε K(36,36) = ε After, the procedure is the same for the other nodes 6, 7 and 8 on the bottom prism. By completing the process for all the nodes located on the top surface, it is assured that all the nodes have the contact with the surface of the top prism. Apparently, the penalty approach does not effect on the DOF in the problem, it means that the penalty coefficient is only a constant value which its amount is changing till reaching the optimal amount. Now the initial penalty coefficient is defined as follows: ε initial = 1e11 Next, the program is run in the state of the penalty method, the following figures demonstrate the deformed mesh related to the penalty method. Figure 16 : Deformed mesh of the second example node to surface with Penalty function method JANUARY 2015 BY JOSÉ BERARDO & BEHZAD FARAHANI 24
26 Apparently, figure 16 strongly proves that there is no penetration and the prisms experienced an acceptable contact. Thus, the assumption of penalty coefficient and force value has the agreement with the applied mathematical relations. Finally, the amount of the penalty coefficient has been updated during the procedure in Matlab and the optimal value is as follows: ε updated = e13 So far, we guessed the dimension of the similar point of nodes located at the bottom prism in the local coordinate system and then the distance was obtained through substituting in the shape function equations and so on. Now, it is going to propose an approach to obtain it automatically in the program. In order to make it clarified, it is better to start with an illustration presented in figure 17. Figure 17: The position of node out of the face and its projection on the face to obtain the shortest distance Referring to figure 17, if a point is considered on the surface as Q on the bottom prism, we can define it: Q(X, Y, Z) Then, the point on the surface of the top prism is assumed as follows: P(P 1, P 2, P 3 ) So, the shortest distance of the node P to the surface (bottom prism) is calculated as; d = (P Q). (P Q) d 2 = (P 1 X) 2 + (P 2 Y) 2 + (P 3 Z) 2 (11) where JANUARY 2015 BY JOSÉ BERARDO & BEHZAD FARAHANI 25
27 X = N 1 x 1 + N 2 x 2 + N 3 x 3 + N 4 x 4 (12-a) Y = N 1 y 1 + N 2 y 2 + N 3 y 3 + N 4 y 4 (12-b) Z = N 1 z 1 + N 2 z 2 + N 3 z 3 + N 4 z 4 (12-c) After that, N 1 = 1 (1 ξ)(1 η) (13-a) 4 N 2 = 1 (1 + ξ)(1 η) (13-b) 4 N 3 = 1 (1 ξ)(1 + η) (13-c) 4 N 4 = 1 (1 + ξ)(1 + η) (13-d) 4 Finally, d 2 = (P 1 X(ξ, η)) 2 + (P 2 Y(ξ, η)) 2 + (P 3 Z(ξ, η)) 2 (14) Now if we rewrite the energy equation in terms of d 2 (it is a function of ξ and η ) and then the derivative in terms of ξ and η are taken: Π ξ = 0 and Π η = 0 After that, ξ and η is obtained associated with the points P and Q. In fact, the distance which was calculated is another definition of the gap between two corresponding nodes at each contact surface of the prisms. The recent manner has been applied to the program in order to calculate the shortest distance (gap) automatically. So, it is only a brief introduction on the equations Further examples of the contact methods In this part, it is focused to present some other benchmarks relevant to node to node and node to surface contact methods. The purpose of this section is relevant to the capability of the Matlab program. Since, the basic equations and concepts were introduced before, the final results are only represented. JANUARY 2015 BY JOSÉ BERARDO & BEHZAD FARAHANI 26
28 First Benchmark - Node to Node Contact, Change in Mesh The material properties are presented in table 1. The new model is created based on the following table. Table 4: The characteristics of the first benchmark, node to node contact Parameter Bottom prism #1 Top prism #2 Dimension L x = L y = L z = 2 [m] L x = L y = L z = 2 [m] Angle of top face slope α 1 = 0 Not Defined Total sum force in one direction chosen in the P 1 = 200 [N] P 2 = 200 [N] top face (z direction) Number of element (N. E) 1 = 27 (N. E) 2 = 27 Number of nodes (N. N) 1 = 64 (N. N) 2 = 64 Gap gp 1 = 0.1 [m] gp 2 = 0.1 [m] (a) (b) Figure 18: The geometry created for first benchmark, (a) Node labeling and (b) Element labeling JANUARY 2015 BY JOSÉ BERARDO & BEHZAD FARAHANI 27
29 Figure 19: Deformed shape of geometry of the first benchmark associated with node to node contact (a) (b) Figure 20: The stress field on the prisms for the first benchmark on (a) Solid body and (b) Cross sections The optimum penalty coefficient is presented below: ε updated = e + 14 JANUARY 2015 BY JOSÉ BERARDO & BEHZAD FARAHANI 28
30 The presented Lagrange coefficient vector is associated with the nodes located at the contact zone. As shown in the vector, the amount of the components are different based on the position of the nodes Second Benchmark - Node to Node Contact, Applying Angle In this part, it is focused on changing the angle of the top face of the bottom prism. It creates a specific slope on the faces located at the bottom prism. The process starts based on the data presented in table 5. The results will be shown for both Penalty and Lagrange multipliers constraints. Table 5: The characteristics of the second benchmark node to node contact Parameter Bottom prism #1 Top prism #2 Dimension L x = L y = L z = 2 [m] L x = L y = L z = 2 [m] Angle of top face slope α 1 = 10 Not Defined Total sum force in one direction chosen in the top P 1 = 400 [N] P 2 = 400 [N] face (z direction) Number of element (N. E) 1 = 8 (N. E) 2 = 8 Number of nodes (N. N) 1 = 27 (N. N) 2 = 27 Gap gp 1 = 0.1 [m] gp 2 = 0.1 [m] JANUARY 2015 BY JOSÉ BERARDO & BEHZAD FARAHANI 29
31 Figure 21 : The geometry created for second benchmark, (a) Node labeling and (b) Element labeling Figure 22 : Deformed shape of geometry of the second benchmark associated with node to node contact JANUARY 2015 BY JOSÉ BERARDO & BEHZAD FARAHANI 30
32 Figure 23: Equivalent von Mises stress on the super convergent points for the second benchmark Then, the penalty coefficient is obtained as: ε updated = e The following Lagrange multipliers coefficient related to this example presented in a vector. The size of the vector is the same as the nodes involved in the contact phenomenon Third Benchmark - Node to surface Contact, Change in Mesh In this section, it is attempted to give the benchmarks related to node to surface contact method associated with the penalty function constraint. The changes are related to the mesh properties and the slope by applying the angle to the faces of the bottom prism. JANUARY 2015 BY JOSÉ BERARDO & BEHZAD FARAHANI 31
33 Table 6: The characteristics of the third benchmark node to surface contact Parameter Bottom prism #1 Top prism #2 Dimension L x = L y = L z = 1 [m] L x = L y = L z = 2 [m] Angle of top face slope α 1 = 0 Not Defined Total sum force in one direction chosen in the top P 1 = 500 [N] P 2 = 500 [N] face (z direction) Number of element (N. E) 1 = 27 (N. E) 2 = 27 Number of nodes (N. N) 1 = 64 (N. N) 2 = 64 Gap gp 1 = 0.05 [m] gp 2 = 0.05 [m] (a) (b) Figure 24 : The geometry created for third benchmark, (a) Node labeling and (b) Element labeling JANUARY 2015 BY JOSÉ BERARDO & BEHZAD FARAHANI 32
34 Figure 25: Deformed shape of geometry of the third benchmark associated with node to surface contact JANUARY 2015 BY JOSÉ BERARDO & BEHZAD FARAHANI 33
35 (a) (b) Figure 26: The displacement field on prisms for the third benchmark on (a) Solid body and (b) Cross sections Then, the penalty coefficient is obtained as: ε updated = e14 Now, it turns to study what happens if the angle applied on the top face of the bottom prism. If the number of elements are 8 and 27 for the bottom and top prisms respectively. Moreover, there are 27 nodes in the bottom prism and 64 nodes in the top prism. The applied force on each prism is P 1 = P 2 = 500 [N]. After that, if the following angle applied on the bottom prism: α 1 = 5 The deformed mesh connected to this model is presented in figure 27. JANUARY 2015 BY JOSÉ BERARDO & BEHZAD FARAHANI 34
36 Figure 27: Deformed shape of geometry of the third benchmark associated with angle = 5 degrees Subsequently; ε updated = e14 JANUARY 2015 BY JOSÉ BERARDO & BEHZAD FARAHANI 35
37 4- Conclusion and discussion First of all, it is noticeable to remark that, in node to node contact phenomenon, the size of two prism ought to be the same. For clarification, let`s give an example with the different prism dimensions. Table 7: The characteristics of the discussion example node to node contact Parameter Bottom prism #1 Top prism #2 Dimension L x = L y = L z = 1 [m] L x = L y = L z = 2 [m] Angle of top face slope α 1 = 0 Not Defined Total sum force in one direction chosen in the top P 1 = 500 [N] P 2 = 500 [N] face (z direction) Number of element (N. E) 1 = 8 (N. E) 2 = 8 Number of nodes (N. N) 1 = 27 (N. N) 2 = 27 Gap gp 1 = 0.05 [m] gp 2 = 0.05 [m] The result of the deformed shape is presented below: Figure 28: Deformed shape of the prisms for node to node contact in the discussion example As seen in figure 28, some results are obtained regarding this procedure. But it is not really convenient. In the node to node contact phenomenon, the nodes should have contact with other exactly. So, in this model the nodes of the bottom prism do not have the opportunity to touch the nodes on the top prism. Although the program is applicable to solve this model, it is a kind of cheating of the node to surface contact phenomenon. JANUARY 2015 BY JOSÉ BERARDO & BEHZAD FARAHANI 36
38 Considering all, it is possible to conclude that: 1- The constraint approaches such as Penalty function and Lagrange Multipliers should be applied in order to prevent the penetration for the bodies. 2- The Lagrange Multiplier coefficient known as λ is a variable which affects the size of the matrices (K, F and U) involved in the solution process. On the other hand, the penalty coefficient ε is characterized as a constant while it is updated during the solving analyses. 3- The amount of applied force could affect strongly the solution of displacement components and Lagrange coefficient. 4- The main advantage of the Lagrange multiplier method is that the impenetrability is satisfied almost perfectly. The main drawback of this approach is the possibility of loss of positive definiteness due to the zero diagonal parts in the discretized governing equations. JANUARY 2015 BY JOSÉ BERARDO & BEHZAD FARAHANI 37
39 Final Remarks The Matlab code corresponding to the current project has the following powerful features: Generating the number of nodes and elements as many as the users define automatically; Predefining the slope of the top face of the bottom prism by the users; Computing the shortest distance between the nodes and their projection on the other face with the ultimate number of nodes automatically; Working with Lagrange Multipliers and Penalty function in node to node contact fully; Working with penalty function in node to surface contact completely; Generating the Von-Mises equivalent stress and displacement field on the body and the cross section and also on the super convergent points. JANUARY 2015 BY JOSÉ BERARDO & BEHZAD FARAHANI 38
40 Reference [1]. Peter Wriggers, Computational Contact Mechanics, Second Edition, University of Hannover, Springer, [2]. Daqing Xu and Keith D. Hjelmstad, A New Node-to-Node Approach to Contact/Impact Problems for Two Dimensional Elastic Solids Subject to Finite Deformation, NSEL Report Series, Report No. NSEL-009, May [3]. Giorgio Zavarise and Laurua de Lorenzis, The node-to-segment Algorithm for 2D Frictionless Contact: Classical Formulation and Special Case Compu. Methods App. Ech Engrg., Elsevier Pub., 198 (2009) [4]. M. Puso, T. Laursen, A Mortar Segment-to-Segment Frictional Contact Method for Large Deformations, Computer Methods in Applied Mechanics and Engineering, October, [5]. Alber Konter, Finite Element Modelling of Contact Phenomena in Structural Analysis, Netherlands Institute for Metals Research, Feb [6]. Kiranraj Shetty, Segment to Segment Contact in Marc, June 2013 JANUARY 2015 BY JOSÉ BERARDO & BEHZAD FARAHANI 39
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