Learning Module 8 Shape Optimization

Learning Module 8 Shape Optimization What is a Learning Module? Title Page Guide A Learning Module (LM) is a structured, concise, and self-sufficient learning resource. An LM provides the learner with the required content in a precise and concise manner, enabling the learner to learn more efficiently and effectively. It has a number of characteristics that distinguish it from a traditional textbook or textbook chapter: An LM is learning objective driven, and its scope is clearly defined and bounded. The module is compact and precise in presentation, and its core material contains only contents essential for achieving the learning objectives. Since an LM is inherently concise, it can be learned relatively quickly and efficiently. An LM is independent and free-standing. Module-based learning is therefore nonsequential and flexible, and can be personalized with ease. Presenting the material in a contained and precise fashion will allow the user to learn effectively, reducing the time and effort spent and ultimately improving the learning experience. This is the first module on structural analysis and covers a static structural study in FEM. It goes through all of the steps necessary to successfully complete an analysis, including geometry creation, material selection, boundary condition specification, meshing, solution, and validation. These steps are first covered conceptually and then worked through directly as they are applied to an example problem. Estimated Learning Time for This Module Estimated learning time for this LM is equivalent to three 50-minute lectures, or one week of study time for a 3 credit hour course. How to Use This Module The learning module is organized in sections. Each section contains a short explanation and a link to where that section can be found. The explanation will give you an idea of what content is in each section. The link will allow you to complete the parts of the module you are interested in, while being able to skip any parts that you might already be familiar with. The modularity of the LM allows for an efficient use of your time. 1

1. Learning Objectives The objective of this module is to introduce the user to the process of Shape Optimization using FEM. Upon completion of the module, the user should have a good understanding of the necessary logical steps of a Shape Optimization simulation, and be able to perform the following tasks: Optimization Key concepts in optimization Objective Design variables State variables Constraints DVs versus SVs A simple example: optimization of a rod Basic and feasible design space Feasible design space Evaluation space An objective in optimization is a parameter of the design that the designer seeks to optimize (i.e., either minimize or maximize). The objective is usually not a parameter the designer can control directly. Instead, it is a function of the design variables that the designer specifies or controls directly. Examples of objective mass, volume, surface area, stress, cost, etc. (for minimization) usable container volume, surface area, natural frequency, etc. (for maximization) You can define only one objective in SW Simulation optimization analysis. The objective is a function of the design variables; that is, changing the values of the design variables leads to change of value of the objective. In fact, this is the very point of optimization change things under your control (design variables) to achieve an objective that is not directly under your control. 2. Prerequisites In order to complete the learning module successfully, the following prerequisites are required: By subject area: o Optimization basics. o Mechanics of Materials or Elasticity By topic: knowledge of o Optimization o Design Variables 2

o State variables o constraints o feasible design o displacement o strain o stress o von Mises stress o Saint Venant s principle o tension, bending, or torsion loading mode 4. Tutorial Problem Statements A good tutorial problem should focus on the logical steps in FEM modeling and demonstrate as many aspects of the FEM software as possible. It should also be simple in mechanics with an analytical solution available for validation. Three tutorial problems are covered in this learning module. Tutorial Problem 1 A rectangular alloy steel plate has one of its ends fixed to the wall. The plate has a width w = 5 mm and a height h = 30 mm. The length of the plate is l = 100 mm. A load of 65 KN acts horizontally on the other end of the plate as shown in the figure. Find the optimal dimensions of the plate using Shape optimization. Figure 1. A Rectangular plate subjected to tensile loading. 3

Tutorial Problem 2 A rectangular alloy steel plate with a hole at its center has one of its ends fixed to the wall. The plate has a width w = 5 mm and a height h = 50 mm. The hole dimensions are shown in the below figure. The length of the plate is l = 100 mm. A tensile load of 46 KN magnitude acts horizontally on the other end of the plate. Find the optimal dimensions of the plate using Shape optimization. Figure 2. A rectangular plate with a hole at its center, rigidly fixed on one end and loaded on the other end 4

Tutorial Problem 3 A rectangular alloy steel plate with a hole at its center has one of its ends fixed to the wall. The plate has a width w = 5 mm and a height h = 50 mm. The hole dimensions are shown in the below figure. The length of the plate is l = 100 mm. A tensile load of 46 KN magnitude acts horizontally on the other end of the plate. Find the optimal dimensions of the plate using Shape optimization. Figure 3. A rectangular plate with a hole at its center rigidly fixed at one end and subjected to bending forces on the other end 5

Pre-test: The pre-test should be taken before taking other sections of the module. The purpose of the pre-test is to assess the user's prior knowledge in subject areas relevant to shape optimization such as Mechanics of Materials and optimization techniques. Questions are focused towards fundamental concepts including stress, strain, displacement, kinematic relationship, constitutive relationship, equilibrium, and material properties, design variables, static variables, constraints, feasible design space, evaluation space. 1. The internal force per unit area acting inside a body when forces are applied to the body is called O Stress O Strain O Displacement O Reaction 2. Shape optimization may result in: O Maximization of Volume O Minimization of Volume O Minimizing mass O All of the above 3. Usage of constraints is of not much importance in shape optimization O True O False 4. Which of the following are not useful in shape optimization. O State Variables O Design Variables O Goals O None of the above 5. Bending moment applied to a beam induces mostly O Tensile stress O Compressive stress O Both tensile and compressive stress O Shear stress 6

6. When the structure is made of same material, which of the following is true? O Minimizing mass and maximizing volume is same. O Minimizing mass and minimizing volume is same. O Mass and Volume are not related O None of the above. 7. For a bar of uniform cross-section under axial loading in x direction, the shape can be optimized by considering the following constraints. O Von-Mises stress, Displacement as variables & mass or volume as goal. O Only volume or mass as goal. O Only Von-Mises Stress, Displacement as variables. O None of the above. 8. For a bar of uniform cross-section under axial loading in x direction, the Young s modulus is equal to O The ratio of the axial displacement to the axial normal stress O The ratio of the x-normal stress to the x-normal strain O The ratio of the xy-shear stress to the x-normal stress O The ratio of the xy-shear stress to the xy-shear strain 9. What is feasible design space? 10. Explain state variables, design variables and constraints 7

Conceptual Analysis of Shape Optimization: Conceptual analysis for a Shape Optimization problem using finite element analysis reveals that the following logical steps and sub-steps are needed: Prerequisite (Associated) Study: 1. Preprocessing 2. Solution 3. Postprocessing 4. Validation Optimization Study: Geometry creation Material property assignment Boundary conditions and loading Mesh generation 1. Preprocessing Objective Design variables Constraints on state variables 2. Solution iterations The above steps are explained in some detail as follows. Prerequisite (Associated) Study: 1. Pre-processing The pre-processing in FEM simulation is analogous to building the structure or making the specimen in physical testing. Several sub-steps involved in pre-processing are geometry creation, material property assignment, boundary condition specification, and mesh generation. The geometry of the structure to be analyzed is defined in the geometry creation step. After the solid geometry is created, the material properties of the solid are specified in the material property assignment step. The material required for the FEM analysis depends on the type of analysis. For example, in the elastic deformation analysis of an isotropic material under isothermal condition, only the modulus of elasticity and the Poisson s ratio are needed. For most novice users of FEM, the boundary condition specification step is probably the most challenging of all pre-processing steps. Two types of boundary conditions are possible. The first is prescribed displacement boundary condition which is analogous to 8

holding or supporting the specimen in physical testing. The second is applied force boundary condition which is analogous to loading the specimen. Several factors contribute to the challenge of applying boundary conditions correctly: 1) Prescribed displacement boundary conditions expressed in terms such as u u const or const are mathematical simplifications, and boundarya x boundaryb frequently only represent supports in real structures approximately. As a result, choosing a good approximate mathematical representation can be a challenge. 2) How a boundary is restrained depends also on the element type. For example, for the "clamped" or "built-in" support, a boundary should be restrained as having zero nodal displacement if solid element is used, while for the same support, the boundary should be restrained as having zero nodal displacement and zero nodal rotation if shell element is used. 3) Frequently, the structure to be analyzed is not fully restrained from rigid body motion in the original problem statement. In order to obtain an FEM solution, auxiliary restraints become necessary. Over-restraining the model, however, leads to spurious stress results. The challenge is then adding auxiliary restraints to eliminate the possibility of rigid body motion without over-restraining the structure. Because of the above challenges, one learning module will be devoted to boundary condition specification. Mesh generation is the process of discretizing the body into finite elements and assembling the discrete elements into an integral structure that approximates the original body. Most FEM packages have their own default meshing parameters to mesh the model and run the analysis while providing ways for the user to refine the mesh. 2. Solution The solution is the process of solving the governing equations resulting from the discretized FEM model. Although the mathematics for the solution process can be quite involved, this step is transparent to the user and is usually as simple as clicking a solution button or issuing the solution command. 3. Post-processing The purpose of an FEM analysis is to obtain wanted results, and this is what the postprocessing step is for. Typically, various components or measures of stress, strain, and displacement at any given location in the structure are available for putout. Additional quantities for output may include factory of safety, energy norm error, contact pressure, reaction force, strain energy density, etc. The way a quantity is outputted depends on the FEM software. 9

Optimization Study: 1. Pre-processing The objective of the optimization study is to get the optimized design for the given model using the FEM package. Select the desired optimization type (Minimize or Maximize), and the desired objective from the dropdown menu (Mass, Volume, Frequency, Buckling) In Response window, select the correct associated study To define a Design Variable: In the Design Variables column select Add parameter. In the Lower Bound box, enter the smallest allowable dimension. In the Upper Bound box, enter the largest allowable dimension. Click OK. To define Constraints on state variables: Select the sensor type like simulation data, mass type and all. Select the appropriate type of sensor. Under Bounds, do the following: Select the desired Units In the Lower Bound box, enter the lowest allowable value for the SV. In the Upper Bound box, enter the highest allowable value for the SV. Click OK. To define goals: Sensor for goal is to be selected from the list of the sensors for mass or volume which is to be maximized or minimized. 10

Overview: In this section, three tutorial problems will be solved using the commercial FEM software SolidWorks. Although the underlying principles and logical steps of an FEM simulation identified in the Conceptual Analysis section are independent of any particular FEM software, the realization of conceptual analysis steps will be software dependent. The SolidWorks-specific steps are described in this section. This is a step-by-step tutorial. However, it is designed such that those who are familiar with the details in a particular step can skip it and go directly into the next step. Tutorial Problem 1. A rectangular beam subjected to tensile loading 0. Launching SolidWorks SolidWorks Simulation is an integral part of the SolidWorks computer aided design software suite. The general user interface of SolidWorks is shown in Figure 1. Main menu Frequently used command icons Help icon Roll over to display File, Tools and other menus Figure 1: general user interface of SolidWorks. In order to perform FEM analysis, it is necessary to enable the FEM component, called SolidWorks Simulation, in the software. Step 1: Enabling SolidWorks Simulation 11

o Click "Tools" in the main menu. Select "Add-ins...". The Add-ins dialog window appears, as shown in Figure 2. o Check the boxes in both the Active Add-ins and Start Up columns corresponding to SolidWorks Simulation. o Checking the Active Add-ins box enables the SolidWorks for the current session. Checking the Start Up box enables the SolidWorks for all future sessions whenever SolidWorks starts up. 1. Pre-Processing Figure 2: Location of the SolidWorks icon and the boxes to be checked for adding it to the panel. Purpose: The purpose of pre-processing is to create an FEM model for use in the next step of the simulation, Solution. It consists of the following sub-steps: Geometry creation Material property assignment Boundary condition specification Mesh generation. 1.1 Geometry Creation The purpose of Geometry Creation is to create a geometrical representation of the solid object or structure to be analyzed in FEM. In SolidWorks such a geometric model is called a part. In this tutorial, the necessary part has already been created in SolidWorks. The following steps will open up the part for use in the FEM analysis. Step 1: Opening the part for simulation. Check SolidWorks Simulation boxes Download the part file tutorial1.sldprt from the web site http://www.femlearning.org/. Use the File menu in SolidWorks to open the downloaded part. 12

The SolidWorks model tree will appear with the given part name at the top. Above the model tree, there should be various tabs labeled Features, Sketch, etc. If the Simulation tab is not visible, go back to steps 1 and 2 to enable the SolidWorks Simulation package. Step 2: Creating a Study o Click the Simulation tab above the model tree o Click on the drop down arrow under Study and select New Study as in Figure 3 o In the Name panel, give the study the name Static Study o Select Static in the Type panel to study the static equilibrium of the part under the load o Click OK to accept and close the menu 1.2 Material Property Assignment Figure 3: The SolidWorks Study menu. The Material Property Assignment sub-step assigns materials to different components of the part to be analyzed. All components must be assigned with appropriate material properties. Step 3: Opening the material property manager o In the upper left hand corner, click Apply Material. o The Material window appears as shown in Figure 4. 13

Figure 4: The Material window. This will apply one material to all components. If the part is made of several components with different materials, open the model tree and apply this process to individual components. 1.3 Boundary Condition Specification In the Boundary Condition Specification sub-step, the restraints and loads on the part are defined. Here, the face of the beam attached to the wall needs to be restrained, and the force in the proper direction needs to be applied on the other end of the beam. Step 5: Opening the fixtures property manager o Right click on Fixtures in the model tree and select Fixed Geometry o Move the cursor into the graphic window. As the cursor traverses the image of the model, notice a small icon accompany the cursor, and this icon change shapes when the cursor is at different locations. This indicates that the SolidWorks is in graphical selection mode, and different shapes indicate different identities would be selected: a square (icon) indicates the surface underneath the cursor will be selected if the mouse is clicked, a line (icon) for an edge or a line, and a dot (icon) for a point. In this tutorial problem, the entire end surface is restrained. 14

Figure 5: Applying an immovable restraint to the beam. At the initial orientation, however, the end to be restrained is not visible, and could not be selected. The model should be rotated to make the fixed end visible. To rotate the model either hold down the scroll bar and rotate with the mouse or change the orientation by clicking on the View Orientation icon in the top middle area of the workspace. Once the desired face is visible, select the face on which to apply the restraint. Note that in the display panel, within the second box in the Type panel, Face<1> appears, indicating that one surface is being selected. Clicking on this face in the graphics panel would deselect the face. Step 6: Restraining the member o Select the face as in Figure 5 o Once the face has been selected, click the green check mark to close the Fixture menu The next step is to load the beam with the applied force. The total force applied is 65000 N in the direction as shown in the figure 6. 15

Step 7: Applying the Tensile Force: Figure 6: Applying the tensile force. o Right click on External Loads in the model tree and select Force. o Under the Force/Torque tab, click the Faces, Edges, Vertices, Reference Points for Force input field box to activate it, if not already active. o Click on the face on which the force is applied in the graphics window. Make sure the face is highlighted (turns blue) and appears in the input field box. o Use SI units and type in a force of 65000 N. o Check the Reverse direction box if the force is pointing in the wrong direction, as shown in Figure 6. o Click OK to close the menu. 1.4 Mesh Generation Purpose: The purpose of the Mesh Generation sub-step is to discretize the part into elements. The mesh consists of a network of these elements. Step 9: Creating the mesh o Right click Mesh in the model tree and select Create mesh o Leave the mesh bar on its default value o Drop down the Advanced menu and make sure the mesh is high quality, not draft quality, by making sure the Draft Quality Mesh checkbox is not clicked o Figure 7 shows the completed mesh o Click OK to close the menu and generate the mesh. 16

Figure 7: A completed mesh. Mesh Control in SolidWorks may be used to refine the mesh locally. The guiding principle is to refine mesh at locations of high stress gradient, such as regions around stress concentrators and locations of geometric changes. For the current problem, local mesh refinement is not pursued. 2. Solution Purpose: The Solution is the step where the computer solves the simulation problem and generates results for use in the Post-Processing step. Step 1: Running the simulation o At the top of the screen, click Run o When the analysis is finished, the Results icon will appear on the model tree 3. Post-Processing Purpose: The purpose of the Post-Processing step is to process the results of interest. For this problem, the von Mises stress and the displacement is of interest. Step 1: Creating a stress plot o Right click Results on the model tree and select Define Stress Plot o Select von Mises as the stress type and Mpa as the unit o Unclick the Deformed Shape box and click OK to close the menu 17

Figure 8: The von Mises stress plot. Step 2: Plotting Displacement plot: o Select the plot for Resultant displacement. Figure 9: The displacement plot. 18

The program finds the optimal solution using many iterations with a Box- Behnken design and displays the initial scenario, optimal scenario, and all iterations. For more information about the quality of the study, see SolidWorks Simulation Help: Properties for the Optimization Design Study. 5. Click Ok. To define the parameter DV1: 1. On the Variable View tab of the Designstudy1 study tab, in the Variables section, select Add Parameter. The Add Parameters dialog box appears. Figure 12: Defining Design Variables. 2. In the Add Parameters dialog box, type DV1 for Name. 3. Set Filter to Model dimensions. 4. In the graphics area, click the D11 dimension shown in the figure. 5. Click OK to close the Add Parameters dialog box. DV1 appears in the Parameters dialog box. 6. Click OK to close the Parameters dialog box.if needed, click to close the Dimension PropertyManager. The parameters DV2 and DV3 are also defined in the same above mentioned procedure. 20

To define the variables: We define the three parameters named DV1, DV2, and DV3 as the variables. 1. On the Variable View tab of the Designstudy1 study tab, under Variables, select DV1 (D11@Sketch1 in the sketch) from the list. The selected variable appears in the Variables section. Figure 13: Defining all Design Variables. 32 2. For the DV1 variable, select Range. Figure 14: Defining type of variable. The program defines the parameter as a continuous variable for optimization. A continuous variable is one which can take any value between the limits. For example, 14.1567mm is a valid value between a minimum value of 25mm and a maximum value of 35mm. 21

3. For Min, type 25mm and for Max, type 35mm. Figure 15: Defining the Minimum and Maximum values for DV. The program varies the model dimension between 25mm and 35mm to find the optimal value for the variable. 4. Repeat steps 1-3 to add the parameter named DV2 (D12@Sketch1 in the sketch) as a variable. Use 95 as minimum and 105 as maximum values for DV2. 5. Repeat steps 1-3 to add the parameter named DV3 (D13@Sketch1 in the sketch) as a variable. Enter 4 and 6 for Min and Max, respectively. The Variables section lists three design variables. Figure 16: Defining the Minimum and Maximum values for all DV s. We need to define sensors to use them as constraints in a Design Study. The Design Study runs the corresponding initial Simulation study to update a sensor's value. For example, it runs the frequency study to track the resonant frequency values. You define a sensor to track the value of the von Mises stress. 1. On the Variable View tab of the Designstudy1 study tab, in the Constraints section, select Add Sensor. 2. In the Property Manager, for Sensor Type, select Simulation Data. 3. Under Data Quantity, for Results, select Stress. 4. For component, select VON: von Mises Stress. 5. Under Properties, for Units, select N/mm^2 (MPa) or Psi. 6. For Criterion, select Model Max and click OK. 7. In the Feature Manager Design tree, under Sensors, rename the sensor as Stress1. 22

Figure 17: Defining the sensors for constraints. 8. Similarly the sensor Displacement1 is also defined. We impose a constraint on the maximum von Mises stress that it should not exceed a value of 620.4 MPa or 89981 Psi. You can use any sensor or driven global variable to define constraints for an Optimization Design Study. For more information about defining sensors as constraints in a Design Study see SolidWorks Simulation Help: Defining Constraints. Figure 18: Defining the Minimum and Maximum values of sensors for constraints. 1. On the Designstudy1 Design Study tab, in the Constraints section, make sure Stress 1 is selected. The sensor appears in the Constraints section. 2. For Mises, select Is less than. 23

3. For Max, type 620.4 N/mm^2. The program automatically selects the Staticstudy study to run and track the sensor's value since only one static study is defined. Defining the Displacement constraint: The maximum resultant displacement should not exceed 0.2056 mm. 1. On the Designstudy1 Design Study tab, in the Constraints section, from the list, select URES. The sensor appears in the Constraints section. This pre-defined sensor tracks the value of resultant displacement. 2. For URES, select Is less than. 3. For Max, type 0.2056 mm. Defining a goal: The program automatically selects the Staticstudy study to run and track the sensor's value. The objective of this Optimization Design Study is to minimize the volume of the part. Figure 19: Defining the Goal. 1. On the Designstudy1 Design Study tab, in the Goals section, from the list, select the Volume1 sensor. 2. The sensor appears in the Goals section. 3. For Volume1, select Minimize. 4. On the Designstudy1 Design Study tab, select Run. 24

Figure 20: Figure showing the run command for iterations. 5. The program runs some iterations (excluding the initial and optimal scenarios) since you defined a High quality study and three design variables. After running the experiments, the program calculates the optimal design variables by forming a response function relating the goal to the variables. Figure 21: Figure showing the process of iterations. Viewing the Results: 1. Review the Initial column. The column is highlighted in red because the constraints on von Mises stress and displacement are violated. 2. Review the Optimal column. The column is highlighted in green because optimization was performed successfully. 25

Figure 22: Figure showing the iterations. The program updates the model with the optimal dimensions in the graphics window. Figure 23: Figure showing the updated models with optimal values. Study Results 15 of 15 iterations ran successfully. Component Units Current Initial Optimal Iteration1 Iteration2 name DV1 mm 30.21393 30 30.21393 35 35 DV2 mm 95.06622 100 95.06622 105 95 DV3 mm 4.9267 5 4.9267 5 5 Stress1 psi 81015 89877 81015 73282 78930 26

Displacement1 mm 0.19697 0.20565 0.19697 0.18508 0.1674 Volume1 mm^3 14151.0696 15000 14151.0696 18375 16625 Component Units Iteration3 Iteration4 Iteration5 Iteration6 Iteration7 name DV1 mm 25 25 35 35 25 DV2 mm 105 95 100 100 100 DV3 mm 5 5 6 4 6 Stress1 psi 97103 1.0274e+005 68618 96292 89803 Displacement1 mm 0.2592 0.23449 0.14682 0.2203 0.20566 Volume1 mm^3 13125 11875 21000 14000 15000 Component Units Iteration8 Iteration9 Iteration10 Iteration11 Iteration12 name DV1 mm 25 30 30 30 30 DV2 mm 100 105 105 95 95 DV3 mm 4 6 4 6 4 Stress1 psi 1.2623e+005 69844 1.0965e+005 77742 1.0874e+005 Displacement1 mm 0.30858 0.17993 0.26998 0.16274 0.24417 Volume1 mm^3 10000 18900 12600 17100 11400 Component name Units Iteration13 DV1 mm 30 DV2 mm 100 DV3 mm 5 Stress1 psi 89877 Displacement1 mm 0.20565 Volume1 mm^3 15000 27

Tutorial Problem 2. A rectangular plate with a hole subjected to tensile loading 3. Launching SolidWorks SolidWorks Simulation is an integral part of the SolidWorks computer aided design software suite. The general user interface of SolidWorks is shown in Figure 1. Main menu Frequently used command icons Help icon Roll over to display File, Tools and other menus Figure 1: general user interface of SolidWorks. In order to perform FEM analysis, it is necessary to enable the FEM component, called SolidWorks Simulation, in the software. Step 1: Enabling SolidWorks Simulation o Click "Tools" in the main menu. Select "Add-ins...". The Add-ins dialog window appears, as shown in Figure 2. o Check the boxes in both the Active Add-ins and Start Up columns corresponding to SolidWorks Simulation. o Checking the Active Add-ins box enables the SolidWorks for the current session. Checking the Start Up box enables the SolidWorks for all future sessions whenever SolidWorks starts up. 28

Check SolidWorks Simulation boxes Figure 2: Location of the SolidWorks icon and the boxes to be checked for adding it to the panel. 1. Pre-Processing Purpose: The purpose of pre-processing is to create an FEM model for use in the next step of the simulation, Solution. It consists of the following sub-steps: Geometry creation Material property assignment Boundary condition specification Mesh generation. 1.1 Geometry Creation The purpose of Geometry Creation is to create a geometrical representation of the solid object or structure to be analyzed in FEM. In SolidWorks such a geometric model is called a part. In this tutorial, the necessary part has already been created in SolidWorks. The following steps will open up the part for use in the FEM analysis. Step 1: Opening the part for simulation. Download the part file tutorial2.sldprt from the web site http://www.femlearning.org/. Use the File menu in SolidWorks to open the downloaded part. The SolidWorks model tree will appear with the given part name at the top. Above the model tree, there should be various tabs labeled Features, Sketch, etc. If the Simulation tab is not visible, go back to steps 1 and 2 to enable the SolidWorks Simulation package. Step 2: Creating a Study o Click the Simulation tab above the model tree 29

o Click on the drop down arrow under Study and select New Study as in Figure 3 o In the Name panel, give the study the name Static Study o Select Static in the Type panel to study the static equilibrium of the part under the load o Click OK to accept and close the menu 1.2 Material Property Assignment Figure 3: The SolidWorks Study menu. The Material Property Assignment sub-step assigns materials to different components of the part to be analyzed. All components must be assigned with appropriate material properties. Step 3: Opening the material property manager o In the upper left hand corner, click Apply Material. o The Material window appears as shown in Figure 4. 30

Figure 5: Applying an immovable restraint to the beam. At the initial orientation, however, the end to be restrained is not visible, and could not be selected. The model should be rotated to make the fixed end visible. To rotate the model either hold down the scroll bar and rotate with the mouse or change the orientation by clicking on the View Orientation icon in the top middle area of the workspace. Once the desired face is visible, select the face on which to apply the restraint. Note that in the display panel, within the second box in the Type panel, Face<1> appears, indicating that one surface is being selected. Clicking on this face in the graphics panel would deselect the face. Step 6: Restraining the member o Select the face as in Figure 5 o Once the face has been selected, click the green check mark to close the Fixture menu The next step is to load the beam with the applied force. The total force applied is 46000 N in the direction as shown in the figure 6. 32

Step 7: Applying the Tensile Force: Figure 6: Applying the tensile force. o Right click on External Loads in the model tree and select Force. o Under the Force/Torque tab, click the Faces, Edges, Vertices, Reference Points for Force input field box to activate it, if not already active. o Click on the face on which the force is applied in the graphics window. Make sure the face is highlighted (turns blue) and appears in the input field box. o Use SI units and type in a force of 46000 N. o Check the Reverse direction box if the force is pointing in the wrong direction, as shown in Figure 6. o Click OK to close the menu. 1.4 Mesh Generation Purpose: The purpose of the Mesh Generation sub-step is to discretize the part into elements. The mesh consists of a network of these elements. Step 9: Creating the mesh o Right click Mesh in the model tree and select Create mesh o Leave the mesh bar on its default value o Drop down the Advanced menu and make sure the mesh is high quality, not draft quality, by making sure the Draft Quality Mesh checkbox is not clicked o Figure 7 shows the completed mesh 33

o Click OK to close the menu and generate the mesh. Figure 7: A completed mesh. Mesh Control in SolidWorks may be used to refine the mesh locally. The guiding principle is to refine mesh at locations of high stress gradient, such as regions around stress concentrators and locations of geometric changes. For the current problem, local mesh refinement is not pursued. 2. Solution Purpose: The Solution is the step where the computer solves the simulation problem and generates results for use in the Post-Processing step. Step 1: Running the simulation o At the top of the screen, click Run o When the analysis is finished, the Results icon will appear on the model tree 3. Post-Processing Purpose: The purpose of the Post-Processing step is to process the results of interest. For this problem, the von Mises stress and the displacement is of interest. Step 1: Creating a stress plot 34

o Right click Results on the model tree and select Define Stress Plot o Select von Mises as the stress type and Mpa as the unit o Unclick the Deformed Shape box and click OK to close the menu Figure 8: The von Mises stress plot. Step 2: Plotting Displacement plot: o Select the plot for Resultant displacement. Figure 9: The displacement plot. 35

Optimization Study: To create an Optimization Design Study: 1. Right-click the Static study tab and select Create New Design Study. Figure 10: Creating News Design Study. 2. Right-click the Design Study tab and select Rename. Rename the study as Designstudy1. 3. On the MinVolume tab, click Design Study Options. 4. In the PropertyManager, under Design Study Quality, select High quality (slower). The program finds the optimal solution using many iterations with a Box- Behnken design and displays the initial scenario, optimal scenario, and all iterations. For more information about the quality of the study, see SolidWorks Simulation Help: Properties for the Optimization Design Study. 5. Click Ok. To define the parameter DV1: 6. On the Variable View tab of the Designstudy1 study tab, in the Variables section, select Add Parameter. The Add Parameters dialog box appears. 36

Figure 11: Defining Design Variables. 7. In the Add Parameters dialog box, type DV1 for Name. 8. Set Filter to Model dimensions. 9. In the graphics area, click the D11 dimension shown in the figure. 10. Click OK to close the Add Parameters dialog box. DV1 appears in the Parameters dialog box. 11. Click OK to close the Parameters dialog box. If needed, click to close the Dimension Property Manager. The parameters DV2 and DV3 are also defined in the same above mentioned procedure. To define the variables: We define the three parameters named DV1, DV2, and DV3 as the variables. 1. On the Variable View tab of the Designstudy1 study tab, under Variables, select DV1 (D11@Sketch1 in the sketch) from the list. The selected variable appears in the Variables section. 2. For the DV1 variable, select Range. Figure 12: Defining type of variable. 37

3. The program defines the parameter as a continuous variable for optimization. A continuous variable is one which can take any value between the limits. For example, 14.1567mm is a valid value between a minimum value of 45mm and a maximum value of 55mm. 4. The program varies the model dimension between 25mm and 35mm to find the optimal value for the variable. 5. Repeat steps 1-3 to add the parameter named DV2 through DV6. The Variables section lists three design variables. Figure 13: Defining the Minimum and Maximum values for all DV s. We need to define sensors to use them as constraints in a Design Study. The Design Study runs the corresponding initial Simulation study to update a sensor's value. For example, it runs the frequency study to track the resonant frequency values. We define a sensor to track the value of the von Mises stress. 1. On the Variable View tab of the Designstudy1 study tab, in the Constraints section, select Add Sensor. 2. In the Property Manager, for Sensor Type, select Simulation Data. 3. Under Data Quantity, for Results, select Stress. 4. For component, select VON: von Mises Stress. 5. Under Properties, for Units, select N/mm^2 (MPa) or Psi. 6. For Criterion, select Model Max and click OK. 7. In the Feature Manager Design tree, under Sensors, rename the sensor as Stress1. 8. Similarly the sensor Displacement1 is also defined as like in the tutorial1. We impose a constraint on the maximum von Mises stress that it should not exceed a value of 620.4 MPa or 89981 Psi. You can use any sensor or driven global variable to define constraints for an Optimization Design Study. For more information about defining sensors as constraints in a Design Study see SolidWorks Simulation Help: 38

Defining Constraints: Figure 14: Defining the Maximum value of sensors for constraints. 1. On the Designstudy1 Design Study tab, in the Constraints section, make sure Stress 1 is selected. The sensor appears in the Constraints section. 2. For Mises, select Is less than. 3. For Max, type 620.4 N/mm^2. The program automatically selects the Staticstudy study to run and track the sensor's value since only one static study is defined. Defining the Displacement constraint: The maximum resultant displacement should not exceed 0.09812 mm. 1. On the Designstudy1 Design Study tab, in the Constraints section, from the list, select URES. The sensor appears in the Constraints section. This pre-defined sensor tracks the value of resultant displacement. 2. For URES, select Is less than. 3. For Max, type 0.09812 mm. Defining a goal: The program automatically selects the Staticstudy study to run and track the sensor's value. The objective of this Optimization Design Study is to minimize the volume of the part. Figure 15: Defining the Goal. 39

1. On the Designstudy1 Design Study tab, in the Goals section, from the list, select the Volume1 sensor. 2. The sensor appears in the Goals section. 3. For Volume1, select Minimize. 4. On the Designstudy1 Design Study tab, select Run. Figure 16: Figure showing the run command for iterations. 5. The program runs some iterations (excluding the initial and optimal scenarios) since you defined a High quality study and three design variables. After running the experiments, the program calculates the optimal design variables by forming a response function relating the goal to the variables. Figure 17: Figure showing the process of iterations. 40

Viewing the Results: 1. Review the Initial column. The column is highlighted in red because the constraints on von Mises stress and displacement are violated. 2. Review the Optimal column. The column is highlighted in green because optimization was performed successfully. Figure 18: Figure showing the iterations. The program updates the model with the optimal dimensions in the graphics window. Figure 19: Figure showing the updated models with optimal values. 41

Study Results 51 of 51 iterations ran successfully. Component Units Current Initial Optimal Iteration1 Iteration2 name DV1 mm 50 53.63739 50 60 60 DV2 mm 100 95.82489 100 110 110 DV3 mm 50 49.86481 50 50 50 DV4 mm 25 24.87213 25 30 20 DV5 mm 5 4.24872 5 5 5 DV6 mm 15 12.73483 15 15 15 Displacement1 mm 0.09812 0.10017 0.09812 0.08779 0.09517 Stress1 N/mm^2 619.37 641.2 619.37 486.52 542.83 (MPa) Volume1 mm^3 24116.42707 21296.37871 24116.42707 32116.42707 32116.42707 Component Units Iteration3 Iteration4 Iteration5 Iteration6 Iteration7 name DV1 mm 60 40 40 40 60 DV2 mm 90 90 90 110 90 DV3 mm 50 50 50 50 50 DV4 mm 20 20 30 30 30 DV5 mm 5 5 5 5 5 DV6 mm 15 15 15 15 15 Displacement1 mm 0.081 0.11553 0.1889 0.20796 0.07325 Stress1 N/mm^2 530.73 832.33 1594.8 1571.1 497.45 (MPa) Volume1 mm^3 26116.42707 17116.42707 17116.42707 21116.42707 26116.42707 Component Units Iteration8 Iteration9 Iteration10 Iteration11 Iteration12 name DV1 mm 40 50 50 50 50 DV2 mm 110 110 110 110 90 DV3 mm 50 60 60 40 40 DV4 mm 20 25 25 25 25 DV5 mm 5 7 3 3 3 DV6 mm 15 15 15 15 15 Displacement1 mm 0.13726 0.07599 0.17742 0.18 0.15098 Stress1 N/mm^2 817.05 437.4 1034.4 1026.6 1033.3 (MPa) Volume1 mm^3 21116.42707 37262.99789 15969.85624 15969.85624 12969.85624 Component Units Iteration13 Iteration14 Iteration15 Iteration16 Iteration17 name DV1 mm 50 50 50 50 50 DV2 mm 90 90 110 90 100 42

DV3 mm 40 60 40 60 60 DV4 mm 25 25 25 25 30 DV5 mm 7 7 7 3 5 DV6 mm 15 15 15 15 20 Displacement1 mm 0.06467 0.06347 0.07711 0.14821 0.12659 Stress1 N/mm^2 438.66 432.71 440.62 1024.2 795.98 (MPa) Volume1 mm^3 30262.99789 30262.99789 37262.99789 12969.85624 23429.20367 Component Units Iteration18 Iteration19 Iteration20 Iteration21 Iteration22 name DV1 mm 50 50 50 50 50 DV2 mm 100 100 100 100 100 DV3 mm 60 60 40 40 40 DV4 mm 30 20 20 20 30 DV5 mm 5 5 5 5 5 DV6 mm 10 10 10 20 20 Displacement1 mm 0.09487 0.09486 0.09366 0.11952 0.11948 Stress1 N/mm^2 567.74 545.79 559.3 793.98 789.36 (MPa) Volume1 mm^3 24607.30092 24607.30092 24607.30092 23429.20367 23429.20367 Component Units Iteration23 Iteration24 Iteration25 Iteration26 Iteration27 name DV1 mm 50 50 60 60 60 DV2 mm 100 100 100 100 100 DV3 mm 60 40 50 50 50 DV4 mm 20 30 30 30 20 DV5 mm 5 5 7 3 3 DV6 mm 20 10 15 15 15 Displacement1 mm 0.1266 0.09366 0.05751 0.13427 0.14682 Stress1 N/mm^2 795.13 547.45 354.55 821.57 896.04 (MPa) Volume1 mm^3 23429.20367 24607.30092 40762.99789 17469.85624 17469.85624 Component Units Iteration28 Iteration29 Iteration30 Iteration31 Iteration32 name DV1 mm 40 40 40 60 40 DV2 mm 100 100 100 100 100 DV3 mm 50 50 50 50 50 DV4 mm 20 20 30 20 30 DV5 mm 3 7 7 7 3 DV6 mm 15 15 15 15 15 Displacement1 mm 0.21067 0.09021 0.14147 0.06288 0.33084 Stress1 N/mm^2 1368.8 586.12 1125.5 382.64 2715.3 (MPa) Volume1 mm^3 11469.85624 26762.99789 26762.99789 40762.99789 11469.85624 Component Units Iteration33 Iteration34 Iteration35 Iteration36 Iteration37 43

name DV1 mm 50 50 50 50 50 DV2 mm 110 110 110 90 90 DV3 mm 50 50 50 50 50 DV4 mm 25 25 25 25 25 DV5 mm 7 7 3 3 3 DV6 mm 20 10 10 10 20 Displacement1 mm 0.08372 0.07168 0.16735 0.13827 0.16655 Stress1 N/mm^2 480.91 399.61 933.51 942.7 1159.1 (MPa) Volume1 mm^3 36300.88514 37950.22129 16264.38055 13264.38055 12557.5222 Component Units Iteration38 Iteration39 Iteration40 Iteration41 Iteration42 name DV1 mm 50 50 50 60 60 DV2 mm 90 110 90 100 100 DV3 mm 50 50 50 60 60 DV4 mm 25 25 25 25 25 DV5 mm 7 3 7 5 5 DV6 mm 20 20 10 20 10 Displacement1 mm 0.07133 0.19545 0.05921 0.09392 0.07733 Stress1 N/mm^2 481.89 1157.8 403.68 573.52 448.27 (MPa) Volume1 mm^3 29300.88514 15557.5222 30950.22129 28429.20367 29607.30092 Component Units Iteration43 Iteration44 Iteration45 Iteration46 Iteration47 name DV1 mm 60 40 40 40 60 DV2 mm 100 100 100 100 100 DV3 mm 40 40 40 60 40 DV4 mm 25 25 25 25 25 DV5 mm 5 5 5 5 5 DV6 mm 10 10 20 20 20 Displacement1 mm 0.07662 0.12207 0.19231 0.20968 0.09177 Stress1 N/mm^2 442.92 733.82 1353.1 1358.2 584.67 (MPa) Volume1 mm^3 29607.30092 19607.30092 18429.20367 18429.20367 28429.20367 Component name Units Iteration48 Iteration49 DV1 mm 40 50 DV2 mm 100 100 DV3 mm 60 50 DV4 mm 25 25 DV5 mm 5 5 DV6 mm 10 15 Displacement1 mm 0.12346 0.09812 Stress1 N/mm^2 (MPa) 734.24 619.37 Volume1 mm^3 19607.30092 24116.42707 44

Tutorial Problem 3. A rectangular plate with a hole subjected to bending loading Launching SolidWorks SolidWorks Simulation is an integral part of the SolidWorks computer aided design software suite. The general user interface of SolidWorks is shown in Figure 1. Main menu Frequently used command icons Help icon Roll over to display File, Tools and other menus Figure 1: general user interface of SolidWorks. In order to perform FEM analysis, it is necessary to enable the FEM component, called SolidWorks Simulation, in the software. Step 1: Enabling SolidWorks Simulation o Click "Tools" in the main menu. Select "Add-ins...". The Add-ins dialog window appears, as shown in Figure 2. o Check the boxes in both the Active Add-ins and Start Up columns corresponding to SolidWorks Simulation. o Checking the Active Add-ins box enables the SolidWorks for the current session. Checking the Start Up box enables the SolidWorks for all future sessions whenever SolidWorks starts up. 45

Check SolidWorks Simulation boxes Figure 2: Location of the SolidWorks icon and the boxes to be checked for adding it to the panel. 1. Pre-Processing Purpose: The purpose of pre-processing is to create an FEM model for use in the next step of the simulation, Solution. It consists of the following sub-steps: Geometry creation Material property assignment Boundary condition specification Mesh generation. 1.1 Geometry Creation The purpose of Geometry Creation is to create a geometrical representation of the solid object or structure to be analyzed in FEM. In SolidWorks such a geometric model is called a part. In this tutorial, the necessary part has already been created in SolidWorks. The following steps will open up the part for use in the FEM analysis. Step 1: Opening the part for simulation. Download the part file tutorial3.sldprt from the web site http://www.femlearning.org/. Use the File menu in SolidWorks to open the downloaded part. The SolidWorks model tree will appear with the given part name at the top. Above the model tree, there should be various tabs labeled Features, Sketch, etc. If the Simulation tab is not visible, go back to steps 1 and 2 to enable the SolidWorks Simulation package. Step 2: Creating a Study o Click the Simulation tab above the model tree 46

Figure 4: The Material window. This will apply one material to all components. If the part is made of several components with different materials, open the model tree and apply this process to individual components. 1.3 Boundary Condition Specification In the Boundary Condition Specification sub-step, the restraints and loads on the part are defined. Here, the face of the beam attached to the wall needs to be restrained, and the force in the proper direction needs to be applied on the other end of the beam. Step 4: Opening the fixtures property manager o Right click on Fixtures in the model tree and select Fixed Geometry o Move the cursor into the graphic window. As the cursor traverses the image of the model, notice a small icon accompany the cursor, and this icon change shapes when the cursor is at different locations. This indicates that the SolidWorks is in graphical selection mode, and different shapes indicate different identities would be selected: a square (icon) indicates the surface underneath the cursor will be selected if the mouse is clicked, a line (icon) for an edge or a line, and a dot (icon) for a point. In this tutorial problem, the entire end surface is restrained. 48

Figure 5: Applying an immovable restraint to the beam. At the initial orientation, however, the end to be restrained is not visible, and could not be selected. The model should be rotated to make the fixed end visible. To rotate the model either hold down the scroll bar and rotate with the mouse or change the orientation by clicking on the View Orientation icon in the top middle area of the workspace. Once the desired face is visible, select the face on which to apply the restraint. Note that in the display panel, within the second box in the Type panel, Face<1> appears, indicating that one surface is being selected. Clicking on this face in the graphics panel would deselect the face. Step 5: Restraining the member o Select the face as in Figure 5 o Once the face has been selected, click the green check mark to close the Fixture menu The next step is to load the beam with the applied bending force. The total force applied is 320 N in the direction as shown in the figure 6. 49

Step 6: Applying the Tensile Force: Figure 6: Applying the tensile force. o Right click on External Loads in the model tree and select Force. o Under the Force/Torque tab, click the Faces, Edges, Vertices, Reference Points for Force input field box to activate it, if not already active. o Click on the face on which the force is applied in the graphics window. Make sure the face is highlighted (turns blue) and appears in the input field box. o Use SI units and type in a force of 320 N. o Check the Reverse direction box if the force is pointing in the wrong direction, as shown in Figure 6. o Click OK to close the menu. 1.4 Mesh Generation Purpose: The purpose of the Mesh Generation sub-step is to discretize the part into elements. The mesh consists of a network of these elements. Step 7: Creating the mesh o Right click Mesh in the model tree and select Create mesh o Leave the mesh bar on its default value o Drop down the Advanced menu and make sure the mesh is high quality, not draft quality, by making sure the Draft Quality Mesh checkbox is not clicked o Figure 7 shows the completed mesh o Click OK to close the menu and generate the mesh. 50

Figure 8: The von Mises stress plot. Step 2: Plotting Displacement plot: o Select the plot for Resultant displacement. Figure 9: The displacement plot. Optimization Study: 52

To create an Optimization Design Study: 1. Right-click the Static study tab and select Create New Design Study. Figure 10: Creating News Design Study. 2. Right-click the Design Study tab and select Rename. Rename the study as Designstudy1. 3. On the Designstudy1 tab, click Design Study Options. 4. In the PropertyManager, under Design Study Quality, select High quality (slower). The program finds the optimal solution using many iterations with a Box- Behnken design and displays the initial scenario, optimal scenario, and all iterations. For more information about the quality of the study, see SolidWorks Simulation Help: Properties for the Optimization Design Study. 5. Click Ok. To define the parameter DV1: 6. On the Variable View tab of the Designstudy1 study tab, in the Variables section, select Add Parameter. The Add Parameters dialog box appears. 53

Figure 11: Defining Design Variables. 7. In the Add Parameters dialog box, type DV1 for Name. 8. Set Filter to Model dimensions. 9. In the graphics area, click the D11 dimension shown in the figure. 10. Click OK to close the Add Parameters dialog box. DV1 appears in the Parameters dialog box. 11. Click OK to close the Parameters dialog box.if needed, click to close the Dimension PropertyManager. The parameters DV2 and DV3 are also defined in the same above mentioned procedure. To define the variables: We define the three parameters named DV1, DV2, and DV3 as the variables. 1. On the Variable View tab of the Designstudy1 study tab, under Variables, select DV1 (D11@Sketch1 in the sketch) from the list. The selected variable appears in the Variables section. 2. For the DV1 variable, select Range. Figure 12: Defining type of variable. 54

3. The program defines the parameter as a continuous variable for optimization. A continuous variable is one which can take any value between the limits. For example, 14.1567mm is a valid value between a minimum value of 45mm and a maximum value of 55mm. 4. The program varies the model dimension between 25mm and 35mm to find the optimal value for the variable. 5. Repeat steps 1-3 to add the parameter named DV2 through DV6. The Variables section lists three design variables. Figure 13: Defining the Minimum and Maximum values for all DV s. We need to define sensors to use them as constraints in a Design Study. The Design Study runs the corresponding initial Simulation study to update a sensor's value. For example, it runs the frequency study to track the resonant frequency values.we define a sensor to track the value of the von Mises stress. 6. On the Variable View tab of the Designstudy1 study tab, in the Constraints section, select Add Sensor. 7. In the Property Manager, for Sensor Type, select Simulation Data. 8. Under Data Quantity, for Results, select Stress. 9. For component, select VON: von Mises Stress. 10. Under Properties, for Units, select N/mm^2 (MPa) or Psi. 11. For Criterion, select Model Max and click OK. 12. In the Feature Manager Design tree, under Sensors, rename the sensor as Stress1. 13. Similarly the sensor Displacement1 is also defined as like in the tutorial1. We impose a constraint on the maximum von Mises stress that it should not exceed a value of 620.4 MPa or 89981 Psi. You can use any sensor or driven global variable to define constraints for an Optimization Design Study. For more information about defining sensors as constraints in a Design Study see SolidWorks Simulation Help: Defining Constraints: Figure 14: Defining the Maximum value of sensors for constraints. 55

1. On the Designstudy1 Design Study tab, in the Constraints section, make sure Stress 1 is selected. The sensor appears in the Constraints section. 2. For Mises, select Is less than. 3. For Max, type 620.4 N/mm^2. The program automatically selects the Staticstudy study to run and track the sensor's value since only one static study is defined. Defining the Displacement constraint: The maximum resultant displacement should not exceed 0.09812 mm. 4. On the Designstudy1 Design Study tab, in the Constraints section, from the list, select URES. The sensor appears in the Constraints section. This pre-defined sensor tracks the value of resultant displacement. 5. For URES, select Is less than. 6. For Max, type 0.09812 mm. Defining a goal: The program automatically selects the Staticstudy study to run and track the sensor's value. The objective of this Optimization Design Study is to minimize the volume of the part. Figure 15: Defining the Goal. 1. On the Designstudy1 Design Study tab, in the Goals section, from the list, select the Volume1 sensor. 2. The sensor appears in the Goals section. 3. For Volume1, select Minimize. 4. On the Designstudy1 Design Study tab, select Run. 56

Figure 16: Figure showing the run command for iterations. 5. The program runs some iterations (excluding the initial and optimal scenarios) since you defined a High quality study and three design variables. After running the experiments, the program calculates the optimal design variables by forming a response function relating the goal to the variables. Viewing the Results: 1. Review the Initial column. The column is highlighted in red because the constraints on von Mises stress and displacement are violated. 2. Review the Optimal column. The column is highlighted in green because optimization was performed successfully. Figure 17: Figure showing the iterations. The program updates the model with the optimal dimensions in the graphics window. 57

And we can plot the results into a word file or html file. And we can list all the iterations as follows: Figure 18: Figure showing the procedure to plot the results into a word file or html file. Study Results 27 of 27 iterations ran successfully. Component Units Current Initial Optimal Iteration1 Iteration2 name DV1 mm 35.20752 40 35.20752 45 45 DV2 mm 251.24512 300 251.24512 350 250 DV3 mm 172.05811 150 172.05811 150 150 DV4 mm 19.99756 15 19.99756 15 15 Stress1 N/mm^2 563.39 616.59 563.39 705.93 435.73 (MPa) Displacement1 mm 22.27108 33.10184 22.27108 46.46544 16.93068 Volume1 mm^3 42658.174 59116.42707 42658.174 77866.42707 55366.42707 Component Units Iteration3 Iteration4 Iteration5 Iteration6 Iteration7 58

name DV1 mm 35 35 40 40 40 DV2 mm 350 250 300 300 300 DV3 mm 150 150 175 175 125 DV4 mm 15 15 20 10 20 Stress1 N/mm^2 1018.1 560.01 594.91 591.27 833.93 (MPa) Displacement1 mm 59.87979 21.93951 33.39551 32.53606 34.44358 Volume1 mm^3 60366.42707 42866.42707 58429.20367 59607.30092 58429.20367 Component Units Iteration8 Iteration9 Iteration10 Iteration11 Iteration12 name DV1 mm 40 45 45 35 35 DV2 mm 300 300 300 300 300 DV3 mm 125 150 150 150 150 DV4 mm 10 20 10 20 10 Stress1 N/mm^2 646.64 602.97 525.23 982.33 675.46 (MPa) Displacement1 mm 32.76191 29.90728 28.99068 39.13601 37.31989 Volume1 mm^3 59607.30092 65929.20367 67107.30092 50929.20367 52107.30092 Component Units Iteration13 Iteration14 Iteration15 Iteration16 Iteration17 name DV1 mm 40 40 40 40 45 DV2 mm 350 350 250 250 300 DV3 mm 175 125 175 125 175 DV4 mm 15 15 15 15 15 Stress1 N/mm^2 709.96 915.71 495.65 512.72 530.29 (MPa) Displacement1 mm 52.00566 52.70332 18.95363 19.3154 29.15408 Volume1 mm^3 69116.42707 69116.42707 49116.42707 49116.42707 66616.42707 Component Units Iteration18 Iteration19 Iteration20 Iteration21 Iteration22 name DV1 mm 45 35 35 40 40 DV2 mm 300 300 300 350 350 DV3 mm 125 175 125 150 150 DV4 mm 15 15 15 20 10 Stress1 N/mm^2 619.47 645.99 902.71 949.97 730.05 (MPa) Displacement1 mm 29.56717 37.63415 38.36122 53.6616 51.53545 Volume1 mm^3 66616.42707 51616.42707 51616.42707 68429.20367 69607.30092 Component Units Iteration23 Iteration24 Iteration25 name DV1 mm 40 40 40 DV2 mm 250 250 300 DV3 mm 150 150 150 DV4 mm 20 10 15 Stress1 N/mm^2 (MPa) 495.13 487.14 616.59 Displacement1 mm 19.46402 18.90288 33.10184 Volume1 mm^3 48429.20367 49607.30092 59116.42707 59

Attachment E. Post-test 1. The internal force per unit area acting inside the body when a force is applied on the body is called: O Stress O Strain O Displacement O Other 2. What is state variable? 3. Define basic and feasible design space 4. What is an Optimum Design? 5. Ranges and limits are set for which entity in an optimization study? O goals O Constraints O both the above O None of the above 6. Bending moment induces: O Tensile stress O Compressive stress O Both tensile and compressive stress O Shear stress 7. State variables can be directly controlled. O True O False 8. A point in the feasible design space represents one feasible design 60

O True O False 9. What is Young s modulus? O The ratio of the normal stress to the normal strain O The ratio of the shear stress to the normal stress O The ratio of the displacement to the normal stress O The ratio of shear stress to shear strain 10. The 2D space in which the horizontal dimension represents all feasible designs and the vertical dimension represents the objective function is: O Feasible design space O Evaluation space O All the above O None of the above 61

Attachment F. Assessment Do you feel it was bad to not have a teacher there to answer any questions you might have? O It didn t matter O It would have been nice O I really wanted to ask a question How did the interactivity of the program affect your learning? O Improved it a lot O Improved it some O No difference O Hurt it some O Hurt it a lot The six levels of Bloom s Taxonomy are listed below. Rank how well this learning module covers each level. 5 meaning exceptionally well and 1 meaning very poor. 1. Knowledge (remembering previously learned material) O 5 O 4 O 3 O 2 O 1 2. Comprehension (the ability to grasp the meaning of the material and give examples) O 5 O 4 O 3 O 2 O 1 3. Application (the ability to use the material in new situations) O 5 O 4 O 3 O 2 O 1 62

4. Analysis (the ability to break down material into its component parts so that its organizational structure may be understood) O 5 O 4 O 3 O 2 O 1 5. Synthesis (the ability to put parts together to form a new whole) O 5 O 4 O 3 O 2 O 1 6. Evaluation (the ability to judge the value of the material for a given purpose) O 5 O 4 O 3 O 2 O 1 Do you think the mixed text and video format works well? O Yes O Indifferent O No Do you think the module presents an affective method of learning FEA? O Yes O Indifferent O No Did you prefer this module over the traditional classroom learning experience? Why or why not. 63

How accurate would it be to call this module self-contained and stand-alone? O Very accurate O Accurate O Indifferent O Inaccurate O Very inaccurate What specifically did you like and/or dislike about the module. How useful were the practice problems? O Very helpful O Helpful O Indifferent O Unhelpful O Very unhelpful Was there any part of the module that you felt was unnecessary of redundant? Was there a need for any additional parts? Please list any suggestions for improving this module. Overall, how would you rate your experience taking this module? O Excellent O Fair O Average O Poor O Awful 64

Attachment G. Practice Problems 1. A rectangular alloy steel plate has one of its ends fixed to the wall. The plate has a width w = 5 mm and a height h = 30 mm. The length of the plate is l = 150 mm. A load of 26800 N force acts horizontally on the other end of the plate as shown in the figure. Find the optimal dimensions of the plate ussssssing Shape optimization. Problem 1 2. A cylindrical rod made up of alloy steel has one of its ends fixed to the wall. The rod has a diameter d = 20 mm. The length of the rod is l = 200 mm. A bending load of 2550 N force acts vertically on the other end of the rod as shown in the figure. Find the optimal dimensions of the rod using Shape optimization. Problem 2 3. A cylindrical rod made up of alloy steel has one of its ends fixed to the wall. The rod has an internal diameter d 1 = 15 mm and an external diameter d 2 = 20 mm. The length 65

of the rod is l = 200 mm. A tensile load of 80000 N force acts along the axis on the other end of the rod as shown in the figure. Find the optimal dimensions of the rod using Shape optimization. Problem 3 4. Follow the tutorial 1 and reverse the direction of loading on the plate. Compare the results obtained with the results obtained in tutorial 1. Explain the difference in results if any. 66