Guide ANSYS Verification Manual: ANSYS Release 9.0

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These zipped archives provide all of the necessary elements for running a test, including geometry parts, material files, and workbench databases. To open a test case in Workbench, locate the archive and import it into Workbench. The results in the databases can be cleared and the tests solved multiple times. The test results should be checked against the verified results in the documentation for each test.

Release Quality Assurance Group. A polygonal area is extruded 60 mm. A rectangular area of 30 mm x 40 mm [having a circular area of radius 6 mm subtracted] is extruded to 20 mm. Both resultant solids form one solid geometry. A rectangular area 24 mm x 5 mm is subtracted from the solid. Two rectangular areas 40 mm x 10 mm are extruded 10 mm and subtracted from solid. Two rectangular areas 25 mm x 40 mm are extruded 40 mm and subtracted from solid.

A Chamfer 10 mm x 10 mm is given to 4 edges on the resultant solid. Four Oval areas are extruded and subtracted from Solid. Fillet Radius 5 mm is given to 4 edges using Blend Feature. Verify Volume of the resultant geometry. A Rectangular area mm x 30 mm is revolved about Z-Axis in to form a Cylinder. A circular area of radius 30 mm is swept mm using Sweep feature. A circular area of radius 30 mm is extruded mm. A solid cylinder is created using Skin-Loft feature between two coaxial circular areas each of radius 30 mm and mm apart.

Volume of Cylinder created after using Skin-Loft feature between two circular areas of Radius 30 mm and mm apart. A rectangular area mm x 88 mm is extruded mm to form a solid box. A solid is subtracted using Skin-Loft feature between two square areas each of side 25 mm and mm apart. The two solid bodies are frozen using Freeze feature. Thus a total of 4 geometries are created. Verify the volume of the resulting geometry. Volume of additional two bodies created due to Sweep feature: Release Force F1 is applied on the face between Parts 2 and 3 and F2 is applied on the face between Parts 1 and 2.

Apply advanced mesh control with element size of 0. Find reaction forces in the Y direction at the fixed supports. Find the Maximum Normal Stress in the x direction on the cylindrical surfaces of the hole. Sizing control with element size of 0. Find the first six modes of natural frequencies.

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Lwo and G. Analysis Type s : Nonlinear Structural Analysis Element Type s : Solid Test Case A cubic shaped body made up of a viscoplastic material obeying Anand's law undergoes uniaxial shear deformation at a constant rate of 0. Find the shear load Fx required to maintain the deformation rate of 0. Find the Temperature Distribution.

The rear face of the heater is insulated. The internal energy generated electrically may be assumed to be uniform and is applied as internal heat generation. Find the maximum temperature and maximum total heat flux. The bar is constrained at both ends by frictionless surfaces.

Advanced mesh control with element size of 2 m is applied. The surface convection coefficient between the spine and the surrounding air is h, the air temper is Ta, and the tip of the spine is insulated. Find the heat conducted by the spine and the temperature of the tip. The multibody is used to nullify the numerical noise near the contact regions. Find the maximum equivalent stress for the whole multibody and the safety factor for each part using the maximum equivalent stress theory with tensile yield limit.

Sizing mesh control with element size of 6. Find the first five modes of natural frequency. The circular edge of the plate is fixed. To get accurate results, apply sizing control with element size of 5 mm on the circular edge. Find the total deformation at the center of the plate. It is axially loaded by two forces: a tensile load at the free end and a compressive load on the flat step face at the junction of the two cross sections.

To get accurate results, apply sizing control with element size of 6. Both the straight edges of the arch are fixed. Find the Load Multiplier for the first buckling mode. The flat end face of the cylinder Shaft 1 is fixed. Harmonic force is applied on the end face of another cylinder Shaft 4 as shown below. Find the z directional Deformation Frequency Response of the system on the face to which force is applied for the frequency range of 0 to Hz for the following scenarios using Mode Superposition. The material of the columns is assigned negligible density so as to make them as massless springs.

The slabs are allowed to move only in the y direction by applying frictionless supports on all the faces of the slabs in the y direction. The end face of the column 2K is fixed and a harmonic force is applied on the face of the slab M as shown in the figure below. Find the y directional Deformation Frequency Response of the system at 70 Hz on each of the vertices as shown below for the frequency range of 0 to Hz using Mode Superposition.

Find the life, damage, and safety factor for the normal stresses in the x, y, and z directions for nonproportional fatigue using the Soderberg theory. Use a design life of 1e6 cycles, a fatigue strength factor or 1, a scale factor of 1, and 1 for coefficients of both the environments under Solution Combination. The fatigue calculations use standard formulae for the Soderberg theory. Other thermal and structural loads are as shown below.

Find the Deformation in the x direction of the contact surface on which the remote force is applied. To get accurate results apply a global element size of 1. Turn on Inertia Relief. Bending loads are applied on the free vertex of the beam as given below. Apply a global element size of 80 mm to get accurate results. Find the deformation in the y direction under Solution Combination with the coefficients for both the environments set to 1.

The density of the longer beam is kept very low so that it acts as a massless spring and the smaller beam acts as a mass. The end vertex of the longer beam acting as a spring is fixed. The cross section details are as shown below. Find the natural frequency of the axial mode. One of the forces is applied on a portion of the beam of length 50 mm L1 from the fixed end and the other is applied on the free vertex, as shown below. Find the load multiplier for the first buckling mode. The end faces of both the parts are fixed and a given displacement is applied on the contact surface of Part 1 as shown below.

Validate all of the above scenarios for Augmented Lagrange and Pure Penalty formulations.

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It is fixed at one end and at the free end a Force F and a Moment M are applied. Use a global element size of 30 mm to get accurate results. See the figure below for details. Find the deformation of the free end in the y direction. A Harmonic force F is applied on the free vertex of the shorter beam in z direction. Both beams have hollow circular cross-sections, as indicated below.

Use both Mode Superposition and Full Method. Both cylinders have length L and both the flat faces of each cylinder are constrained in the axial direction. They are free to move in radial and tangential directions. An internal pressure of P is applied on the inner surface of the inner cylinder. To get accurate results, apply a global element size of 0.

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Find the maximum tangential stresses in both cylinders. Note Tangential stresses can be obtained in the Mechanical application using a cylindrical coordinate system. Use the x-stress component. Consider load type as fully reversed and a Design Life of 1e6 cycles, Fatigue Strength factor of 1, and Scale factor of 1. Find the maximum temperature, maximum total heat flux, maximum total deformation, and heat reaction at the given temperature.

The longitudinal faces have frictionless support. A bolt pretension load is applied on the semi-cylindrical face. To get accurate results, apply sizing control with element size of 0. Find the Z directional deformation and the adjustment reaction due to the bolt pretension load. To get accurate results, set the advanced mesh control element size to 0.

Find X normal stress at a distance of 0. Also find total deformation and reaction moment. Analysis takes into account the unit length in the z-direction. So is there a way to run these test cases? Permalink 0 0 0. Assuming you are working on a Windows OS, copy the vm Use -np 2 at the end if you have only 2 cores. Search in Employing Birth and Death Define the First Load Step Sample Input for First Load Step Define Subsequent Load Steps Sample Input for Subsequent Load Steps Sample Input for Deactivating Elements User-Programmable Features and Nonstandard Uses User-Programmable Features Employing User-Programmable Features Types of User-Programmable Features Available What Are Nonstandard Uses?

Activating Parallel Processing System-Specific Considerations Index—1 List of Figures 1. Example of a Beam for Design Optimization Optimization Data Flow Local and Global Minima Beam With Two Load Cases History of Objective and Constraint Functions Final Topological Shape for Second Scenario Two-Story Planar Frame History of Fundamental Frequency A Beam Under a Snow Load Probabilistic Design Data Flow Histograms for the Snow Height H1 and H Cumulative Distribution Function of X Range of Scatter Effects of Reducing and Shifting Range of Scatter Element Plot for Waveguide Example Graph of Phase Angle Graph of Magnitude Basic Sector Definition Examples of Nodal Diameters i Traveling Wave Animation Example Contour Line Plot of Equipotentials Submodeling of a Pulley Coarse Model Submodel Superimposed Over Coarse Model Cut Boundaries on the Submodel Loads on the Submodel Contour Plots to Compare Results Path Plots to Compare Results Node Rotations Applicable Solvers in a Typical Substructuring Analysis Example of a Substructuring Application Node Locations Connecting a Structure to an Interface Point Linkage Assembly Link3 Component Valid Non-Cyclically Symmetric Loads Loads Applicable in a Substructure Analysis Chapter 1: Design Optimization The ANSYS program can determine an optimum design, a design that meets all specified requirements yet demands a minimum in terms of expenses such as such as weight, surface area, volume, stress, cost, and other factors.

An optimum design is one that is as effective as possible. Virtually any aspect of your design can be optimized: dimensions such as thickness , shape such as fillet radii , placement of supports, cost of fabrication, natural frequency, material property, and so on. The following design optimization topics are available: 1. Getting Started with Design Optimization 1. Optimizing a Design 1.

Multiple Optimization Executions 1. Optimization Mehods 1. Guidelines for Choosing Optimization Variables 1. Hints for Performing Design Optimization 1. Sample Optimization Analysis 1. Getting Started with Design Optimization This section introduces you to design optimization terminology and information flow. Upper and lower limits are specified to serve as "constraints" on the design variables. These limits define the range of variation for the DV.

In the above beam example, width b and height h are obvious candidates for DVs. Also, h has an upper limit of hmax. Also known as "dependent variables," they are typically response quantities that are functions of the design variables. A state variable may have a maximum and minimum limit, or it may be "single sided," having only one limit. It should be a function of the DVs that is, changing the values of the DVs should change the value of the objective function.

In the beam example, the total weight of the beam could be the objective function to be minimized. You must identify which parameters in the model are DVs, which are SVs, and which is the objective function. Typically, a design set is characterized by the optimization variable values; however, all model parameters including those not identified as optimization variables are included in the set.

If any one of the constraints is not satisfied, the design is considered infeasible. The best design is the one which satisfies all constraints and produces the minimum objective function value. If all design sets are infeasible, the best design set is the one closest to being feasible, irrespective of its objective function value.

The file must contain a parametrically defined model, using parameters to represent all inputs and outputs to be used as DVs, SVs, and the objective function. LOOP , created automatically via the analysis file. The design optimizer uses the loop file to perform analysis loops. Output for the last loop performed is saved in file Jobname. An or simply iteration is optimization iteration One or more analysis loops which result in a new design set.

Typically, an iteration equates to one loop; however, for the first order method, one iteration represents more than one loop. This database can be saved to Jobname. OPT or resumed at any time in the optimizer. Section 1. Information Flow for an Optimization Analysis The following figure illustrates the flow of information during an optimization analysis.

Figure 1. This may be a more efficient method for complex analyses for example, nonlinear that require extensive run time. When performing optimization through the GUI, it is important to first establish the analysis file for your model. Then all operations within the optimizer can be performed interactively, allowing the freedom to probe your design space before the actual optimization is done. The insights you gain from your initial investigations can help to narrow your design space and achieve greater efficiency during the optimization process.

The interactive features can also be used to process batch optimization results. General Process for Design Optimization The process involved in design optimization consists of the following general steps. The steps may vary slightly, depending on whether you are performing optimization interactively through the GUI , in batch mode, or across multiple machines. Create an analysis file to be used during looping. Declare optimization variables OPT. Choose optimization tool or method OPT. Specify optimization looping controls OPT.

Initiate optimization analysis OPT.

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  5. Details of the optimization process are presented below. Differences in the procedure for a "batch" versus "interactive" approach are indicated, where appropriate. The program uses the analysis file to form the loop file, which is used to perform analysis loops. In this file, the model must be defined in terms of parameters which are usually the DVs , and results data must be retrieved in terms of parameters for SVs and the objective function.

    Only numerical scalar parameters are used by the design optimizer. See Section 5. It is your responsibility to create the analysis file and to verify that it is correct and complete. It must represent a clean analysis that will run from start to finish. Most nonessential commands such as those that perform graphic displays, listings, status requests, etc.

    Keep in mind that the analysis file will be used over and over again during optimization looping. Any commands that do not have direct bearing on the analysis will produce wasted action and therefore decrease looping efficiency. Both methods have advantages and disadvantages. Creating the file with a system editor is the same as creating a batch input file for the analysis. If you are performing the entire optimization in batch mode, the analysis file will usually be the first portion of the complete batch input stream.

    This method allows you full control of parametric definitions through exact command inputs. It also eliminates the need to clean out unnecessary commands later. You may find it easier to perform the initial analysis interactively, and then use the resulting command log as the basis for the analysis file. In this case, final editing of the log file may be required in order to make it suitable for optimization looping. See Section 1. No matter how you intend to create the analysis file, the basic information that it must contain is the same.

    The steps it must include are explained next. For our beam example, the DV parameters are B width and H height , so the element real constants are expressed in terms of B and H Initialize width! Initialize height! Young's modulus! Leave PREP7 As mentioned earlier, you can vary virtually any aspect of the design: dimensions, shape, material property, support placement, applied loads, etc.

    The only requirement is that the design must be defined in terms of parameters. The initial values assigned to these parameters represent a starting design, which is later modified by the optimizer. However, some picking operations do not allow parametric input. Therefore, you should avoid these picking operations when defining items that will be used as DVs, SVs, or an objective function. Instead, use menu options that allow direct input of parameters. All data required for the analysis should be specified: master degrees of freedom in a reduced analysis, appropriate convergence criteria for nonlinear analyses, frequency range for harmonic response analysis, and so on.

    Loads and boundary conditions may also be DVs. Static analysis default! Transverse pressure load per unit! You can, for instance, obtain a thermal solution and then obtain a stress solution for thermal stress calculations. If your solution uses the multiframe restart feature, all changes to the parameter set that are made after the first load step will be lost in a multiframe restart.

    Retrieve Results Parametrically This is where you retrieve results data and assign them to parameters.

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    These parameters usually represent SVs and the objective function. POST1 is typically used for this step, especially if the data are to be stored, summed, or otherwise manipulated. In our beam example, the weight of the beam is the objective function to be minimized. Because weight is directly proportional to volume, and assuming uniform density, minimizing the total volume of the beam is the same as minimizing its weight. Therefore, we can use volume as the objective function. The SVs for this example are the total stress and deflection. The parameters for these data may be defined as follows Sums the data in each column of the element!

    Sorts nodes based on UY deflection! Preparing the Analysis File So far, we have described what needs to go into the analysis file. If you create this file as you would a batch input file entering commands with a system editor , then you simply save the file and begin the optimization procedure see Section 1. However, if you choose to create your model interactively in ANSYS, you must derive the analysis file from the interactive session.

    This can be done one of two ways, using the database command log or the session log file. The internal log contains all commands that were used to create the current database. Session Log File - Jobname. LOG contains all commands that are issued during an interactive session. To use Jobname. LOG as the optimization analysis file, you should edit out all nonessential commands with a system editor. Because all commands issued are saved to this file, extensive editing may be needed. Also, if your analysis was performed in several ANSYS sessions, you should piece together the log file segments for each session to create a complete analysis file.

    For more details on creating this file, see Section 1. Establish Parameters for Optimization At this point, having completed the analysis file, you are ready to begin optimization. When performing optimization interactively, it is advantageous but optional to first establish the parameters from the analysis file in the ANSYS database. This step is not necessary for optimization performed in batch mode. There are two possible reasons for performing this step. The initial parameter values may be required as a starting point for a first order optimization.

    Also, for any type of optimization run, having the parameters in the database makes them easy to access through the GUI when defining optimization variables. DB associated with the analysis file. Doing this requires that you know which parameters were defined in the analysis file. Note — The ANSYS database does not need to contain model information corresponding to the analysis file to perform optimization.

    The model input will be read from the analysis file automatically during optimization looping. When you first enter the optimizer, any parameters that exist in the ANSYS database are automatically established as design set number 1. It is assumed that these parameter values represent a potential design solution. The file is used to derive the optimization loop file Jobname. There is no default for the analysis file name, therefore it must be input. In batch mode the analysis file name defaults to Jobname.

    BAT a temporary file containing input copied from the batch input file. Therefore, you normally do not need to specify an analysis file name in batch mode. This is required for proper loop file construction. Declare Optimization Variables The next step is to declare optimization variables, that is, specify which parameters are DVs, which ones are SVs, and which one is the objective function.

    As mentioned earlier, up to 60 DVs and up to SVs are allowed, but only one objective function is allowed. No constraints are needed for the objective function. Each variable has a tolerance value associated with it, which you may input or let default to a program calculated value. You may change a previously declared optimization variable at any time by simply redefining it.

    The delete option does not delete the parameter; it simply deactivates the parameter as an optimization variable see Section 1. Single loop is the default. Two methods are available: the subproblem approximation method and the first order method. In addition, you can supply an external optimization algorithm, in which case the ANSYS algorithm will be bypassed. To use one of these methods, you must have an objective function defined. It is a general method that can be applied efficiently to a wide range of engineering problems.

    It is highly accurate and works well for problems having dependent variables that vary widely over a large range of design space. However, this method can be computationally intense. Optimization tools are techniques used to measure and understand the design space of your problem. Since minimization may or may not be a goal, an objective function is not required for use of the tools.

    However, design variables must be defined. The following tools are available. You can do "what if" studies by using a series of single loop runs, setting different design variable values before each loop. A maximum number of loops and a desired number of feasible loops can be specified. This tool is useful for studying the overall design space, and for establishing feasible design sets for subsequent optimization analysis. Specifically, it varies one design variable at a time over its full range using uniform design variable increments. This tool makes global variational evaluations of the objective function and of the state variables possible.

    This technique is related to the technology known as design of experiment that uses a 2-level, full and fractional factorial analysis. The primary aim is to compute main and interaction effects for the objective function and the state variables. Using this tool, you can investigate local design sensitivities. Specify Optimization Looping Controls Each method and tool has certain looping controls associated with it, such as maximum number of iterations, etc. There are also a number of general controls which affect how data is saved during optimization. In addition, you can specify which type of parameters are to be saved during looping: scalar parameters only default , or scalar and array parameters.

    This capability allows for control of parameter values DV and non-DV during looping. LOOP is written from the analysis file. This loop file, which is transparent to the user, is used by the optimizer to perform analysis loops. Looping will continue until convergence, termination not converged, but maximum loop limit or maximum sequential infeasible solutions encountered , or completion for example, requested number of loops for random design generation has been reached.

    If a loop is interrupted due to a problem within the model for example, a meshing failure, a non-converged nonlinear solution, conflicting design variable values, etc. In interactive mode, a warning will be issued, and you may choose to continue or terminate looping. In batch mode, looping will automatically continue. The design set for the aborted loop will be saved, but the response quantity parameters for that set will have very large, irrelevant values.

    The values of all optimization variables and other parameters at the end of each iteration are stored on the optimization data file Jobname. Up to such sets are stored. When the th set is encountered, the data associated with the "worst" design are discarded. Analysis file name not needed for batch! Initiate optimization looping Several optimization executions may occur in series. For example, we could perform a sweep generation after the subproblem approximation execution is completed. Sweep evaluation tool! Review Design Sets Data After optimization looping is complete, you can review the resulting design sets in a variety of ways using the commands described in this section.

    These commands can be applied to the results from any optimization method or tool. There are several specialized ways to review results from the sweep, factorial, and gradient tools. Menu paths appear in detailed discussions of these commands later in this chapter. When issued within the optimizer, this command lists other current optimization information such as the analysis file name, specified optimization technique, number of existing design sets, optimization variables, etc. Using the STAT command is a good way to check the current optimization environment at any point in the optimizer and to verify that all desired settings have been input.

    By default, results are saved for the last design set in file Jobname. RST or. RTH, etc.

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    The "best results" will be in file Jobname. BRTH, etc. Manipulating Designs Sets After reviewing the design sets, it may be desirable to manipulate them in some way. For example, after performing a random design execution, you may wish to discard all non-feasible designs, keeping the feasible sets as data points for subsequent optimization.

    There are several ways in which you can change the design sets. Two commands are available for discarding unwanted sets. Up to design sets can be stored in the optimization database. There are other commands that can affect design sets. Multiple Optimization Executions There are various reasons why you might wish to do more than one optimization execution.

    For example, your initial optimization run may not find the desired optimum. Or, you may start by using a design tool and then perform a subsequent optimization for example, random design generation followed by a subproblem approximation run. The knowledge you gain from the first few runs may prompt you to change your design space and optimize yet again. If you perform all executions within the same ANSYS session or within the same batch input stream , the procedure is very straightforward.

    After an execution, simply redefine all optimization input as desired and initiate the next execution. Restarting an Optimization Analysis To restart an optimization analysis, simply resume the optimization database file Jobname. The analysis file corresponding to the resumed database must be available in order to perform optimization.

    Read named file defaults to Jobname. If there is data in the optimization database at the time you want to resume, you should first clear the optimization database. When you do this, all settings are reset to their default values, and all design sets are deleted. For both the subproblem approximation and first order methods, the program performs a series of analysisevaluation-modification cycles. The process is repeated until all specified criteria are met. In addition to the two optimization techniques, the ANSYS program offers a set of strategic tools that can be used to enhance the efficiency of the design process.

    For example, a number of random design iterations can be performed. The initial data points from the random design calculations can serve as starting points to feed the optimization methods described. The following topics about design optimization methods are available: 1.

    Subproblem Approximation Method 1. First Order Method 1. Random Design Generation 1. Using the Sweep Tool 1. Using the Factorial Tool 1. Theory Reference. Subproblem Approximation Method The subproblem approximation method can be described as an advanced zero-order method in that it requires only the values of the dependent variables, and not their derivatives. There are two concepts that play a key role in the subproblem approximation method: the use of approximations for the objective function and state variables, and the conversion of the constrained optimization problem to an unconstrained problem.

    Approximations For this method, the program establishes the relationship between the objective function and the DVs by curve fitting. This is done by calculating the objective function for several sets of DV values that is, for several designs and performing a least squares fit between the data points. The resulting curve or surface is called an approximation. Each optimization loop generates a new data point, and the objective function approximation is updated. It is this approximation that is minimized instead of the actual objective function.

    State variables are handled in the same manner. An approximation is generated for each state variable and updated at the end of each loop. You can control curve fitting for the optimization approximations. You can request a linear fit, quadratic fit, or quadratic plus cross terms fit. By default, a quadratic plus cross terms fit is used for the objective function, and a quadratic fit is used for the SVs. Theory Reference for details.

    Conversion to an Unconstrained Problem State variables and limits on design variables are used to constrain the design and make the optimization problem a constrained one. The ANSYS program converts this problem to an unconstrained optimization problem because minimization techniques for the latter are more efficient. The conversion is done by adding penalties to the objective function approximation to account for the imposed constraints. The search for a minimum of the unconstrained objective function approximation is then carried out by applying a Sequential Unconstrained Minimization Technique SUMT at each iteration.

    Convergence Checking At the end of each loop, a check for convergence or termination is made. It only means that one of the four criteria mentioned above has been satisfied. Therefore, it is your responsibility to determine if the design has been sufficiently optimized. If not, you can perform additional optimization analyses.

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    Sometimes the solution may terminate before convergence is reached. The default number is 7. Special Considerations for Subproblem Approximation Because approximations are used for the objective function and SVs, the optimum design will be only as good as the approximations. Guidelines to help establish "good" approximations are presented below. For subproblem approximation, the optimizer initially generates random designs to establish the state variable and objective function approximations. Because these are random designs, convergence may be slow.

    You can sometimes speed up convergence by providing more than one feasible starting design. Simply run a number of random designs and discard all infeasible designs. If many infeasible designs are being generated by the subproblem approximation method, it may mean that the state variable approximation does not adequately represent the actual state variable function. First Order Method Like the subproblem approximation method, the first order method converts the problem to an unconstrained one by adding penalty functions to the objective function. However, unlike the subproblem approximation method, the actual finite element representation is minimized and not an approximation.

    The first order method uses gradients of the dependent variables with respect to the design variables. For each iteration, gradient calculations which may employ a steepest descent or conjugate direction method are performed in order to determine a search direction, and a line search strategy is adopted to minimize the unconstrained problem. Thus, each iteration is composed of a number of subiterations that include search direction and gradient computations.

    That is why one optimization iteration for the first order method performs several analysis loops. Typically, the default values for these two inputs are sufficient. Convergence Checking First order iterations continue until either convergence is achieved or termination occurs. It is also a requirement that the final iteration used a steepest descent search, otherwise additional iterations are performed.

    Special Considerations for the First Order Method Compared to the subproblem approximation method, the first order method is seen to be more computationally demanding and more accurate. However, high accuracy does not always guarantee the best solution. In this case, it has probably found a local minimum, or there is no feasible design space. Also, you may try generating random designs OPTYPE,RAND to locate feasible design space if any exists , then rerun the first order method using a feasible design set as a starting point.

    This is because first order starts from one existing point in design space and works its way to the minimum. If the starting point is too near a local minimum, it may find that point instead of the global minimum. If you suspect that a local minimum has been found, you may try using the subproblem approximation method or random design generation, as described above. Because this method solves the actual finite element representation not an approximation , it will strive to find an exact solution based on the given tolerance.

    If a number of feasible design sets is specified, looping will terminate when that number is reached, even if the maximum number of iterations has not been reached. Random design generation is often used as a precursor to a subproblem approximation optimization as explained earlier. It can also be used to develop trial designs for a variety of purposes.

    For example, a number of random designs can be generated, then reviewed in order to judge the validity of the current design space. The DV in question is incremented for each loop, while the remaining design variables are held fixed at their reference values. The horizontal axis shows normalized values 0 to 1 for the design variables, where the normalization is with respect to the DV maximum and minimum values OPVAR. For the design variables, 0 corresponds to its minimum value and 1 to its maximum. The 2-level technique samples the 2 extreme points of each DV. For a full evaluation, the program performs 2n loops, where n is the number of design variables.

    You can display graphics in the form of bar charts and generate tables that show certain effects for either the objective function or any state variable. For example, you may request a graph of the main effect that each design variable has on the objective function.

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    You can also see the effects associated with 2- and 3-variable interactions. Theory Reference for more information. Gradient results are useful for studying the sensitivities of the objective function or state variables. You can graph response variables with respect to design variable values. The vertical axis shows actual values for the objective function or state variable graphed. Some of these are presented below. Choosing Design Variables DVs are usually geometric parameters such as length, thickness, diameter, or model coordinates.

    They are restricted to positive values. Having too many design variables increases the chance of converging to a local minimum rather than the true global minimum, or even diverging if your problem is highly nonlinear. Obviously, more DVs demand more iterations and, therefore, more computer time.

    One way to reduce the number of design variables is to eliminate some DVs by expressing them in terms of others, commonly referred to as design variable linking. DV linking may not be practical if the DVs are truly independent. However, it may be possible to make judgements about your structure's behavior which allow a logical link between some DVs. For example, if it is thought that an optimum shape will be symmetric, use one DV for symmetric members. Too wide a range may result in poor representation of design space, whereas too narrow a range may exclude "good" designs. Remember that only positive values are allowed, and that an upper limit must be specified.