Chapter 8 Visualization and Optimization

Transcription

1 Chapter 8 Visualization and Optimization Recommended reference books: [1] Edited by R. S. Gallagher: Computer Visualization, Graphics Techniques for Scientific and Engineering Analysis by CRC, 1994 [2] Edited by L. Rosenblum and et al: Scientific Visualization, Advances and Challenges by Academic Press, 1994 [3] D. Pieere: Optimization Theory with Applications, by Dover Publications, INC, 1986 [4] G.Winter and et al: Genetic Algorithm in Engineering and Computer Science, by John Wiley & Sons, Computer Visualization Introduction Computer graphics visualization of behavior is a young field witch has grown to have a major impact on the practice of engineering design and analysis. Visualization is defined in the dictionary as a mental image. In the field of computer graphics and engineering design the term has a much more specific meaning: The technical specialty of visualization concerns itself with the display of behavior, and particularly, with making complex states of behavior comprehensible to the human eyes Scientific Visualization Computer graphics visualization techniques have formed a natural marriage with computer simulations of behavior in understanding the physical phenomena of engineering problems. A. Computer Aided Engineering: a. Computational methods for analysis Such as FEM, FDM, MOM and BEM. Digital numerical analysis techniques generally shared the common feature of producing numerical results at point in space, representable by polygonal, polyhedral or higher-order geometry. The need to see and understand this data, combine with its readily accessible geometric form, made numerical methods an ideal early application for the engineering field of computer graphics. b. Computer graphics and visualization Newer 3-D visualization techniques allowing the display of volumetric, multivariate and time-dependent behavior have joined earlier display techniques in numerical 1

2 analysis applications. In addition to the visualization algorithms themselves, this trend has been driven by increasing computing capabilities, which allow for more complex analyses. Today, the move towards further real-time 3-D graphics display capabilities is helping visualization become part of a more interactive approach to analysis. B. The Process of Analysis and Visualization Many engineering design and analysis applications share common components of modeling, analysis, and the visualization of results. The first and the last of these are often described in engineering practice as pre-processing and post-processing, respectively, as functions which take place before and after analysis. Fig. 8.1 describes the traditional sequential design process. Fig. 8.1 The traditional sequential design process. There has been an increasing trend toward combing these functions within an interactive analysis laboratory, outline in Fig In such an environment, the analysis itself may govern geometric changes, and the results can be visualized as the analysis proceeds. Within this framework, one can still describe the individual components of this analysis cycle as modeling, analysis and visualization activities. Fig. 8 Outline of combing three functions within an interactive analysis laboratory. 2

3 Steps involved in modeling include operations such as: Geometric Modeling. An analysis project often begins with construction of a geometric model describing the problem. Analysis Modeling. This geometric model is used to create the date needed to perform an analysis. Generation of Initial Conditions. The same analysis model can conceivably be used to solve for a number of states of behavior: different loads, initial states, or boundary conditions, for example. There has been a clear trend over time towards interactive, graphically oriented methods for specifying this transient analysis data, and towards modifying this data directly within the analysis process itself. Once a model has been created and its conditions specified, an analysis can be performed. Upon completion of this analysis, or increasingly at intermediate stages of analysis, results are available to be visualized in the post-processing step. While the specific kinds of visualization techniques used in post-processing vary as widely as analysis applications themselves, some of its more common operations including the follows: Result Rendering. Techniques such as contour displays, isosurfaces, tensor field streamlines and many others are employed to create images representing the behavior of a scalar vector, tensor or multivariate state of behavior. Model Transformation. The more complex the models, the more important it becomes to be able to interactively modify the view of the model, as well as other display attributes such as perspective, distance, and the position of the observer. Animation. A great deal of actual behavior involves motion, or variations over time. In result visualization, animation involves two principal issues: the display of behavior over time; and navigating or walking through a model in 3-D space. Result Field Manipulation. Beyond the animation of a fixed, known result quantity, newer user interface and display techniques make it possible to interactively explore the nature of a result field. Such techniques include probing 3-D locations within a model to display field data at a given point, interactive display of isovalues and result value levels, volume clipping and slicing, and the rapid comparison of multiple states of behavior. Modeling, analysis and visualization form an analogy to the physical process of engineering design. Its early development closely followed the existing practice of product design, modeling, testing, and changing. With greater interactive computer resources, it is now becoming easier to go beyond the conceptual limitations of physical design, and simultaneously design, test, and modify objects by computing simulation. 3

4 C. The Impact of Visualization in Engineering Design Some of the ways visualization techniques have affected engineering practice today include: Decreased Physical Testing. Analysis and visualization generally provide a simulation of something in real life. This testing on the computer translates to much less physical testing, often at substantial cost savings. Greater Integration of Design and Analysis. The better one can visualize analysis results, the easier it is to modify a design to correct flaws shown by the analysis. There has been a gradual evolution in engineering practice towards doing what if studies at an earlier stage of the design process. Increased Analysis Complexity and Sophistication. Digital numerical analysis has always been limited by the ability to visualize its results. Color graphics displays have had a direct impact on problem size and complexity, and newer 3-D visualization techniques in particular have made it possible to examine complex phenomena which are not readily seen on the exterior visible surfaces of an analysis model. Design Optimization. By using analysis results to parametric attributes of a model, such as length or shape values, and analysis can be used to automatically guide design changes. This has helped analysis evolve from a tool for evaluating static designs to a means of optimizing the design itself. Visualization tools play a key role in this process, allowing engineers to evaluate automated design changes and analysis results. Productivity and Accuracy, Improved visualization tools help relieve a perceptional bottleneck in the man-time spent in the analysis cycle, allowing a faster and more through design process. In addition, more accurate control of the result data seen by the engineer can lead to better design decisions. Beyond the technical advantages produced by computer graphic visualization techniques, there is a clear economic benefit in design productivity Computer Graphics for Visualization Computer graphics has developed into a reasonably complex subject, it will be impossible in this lecture to reference every topic. The basic concepts like 3D Computer- Generated Images, Modeling, Projection, Rendering and Animation will be introduced in this section. A. What is Computer Graphics? 4

5 The simple answer is: Computer Graphics is making pictures with computer. In fact, we could use the aggregate content of theses, papers, and communications in journal and conferences to define the area. This will help, but, as with any definition, there will be grey areas, with work at the boundary between computer graphics and computer vision, image processing, human factors, various engineering disciplines especially electrical engineering, and even mathematics. A. Why Computer Graphics? It is endless to list all the areas of application of computer graphics, as indicated above, but if we make a short list of applications where computer graphics plays an important role, it could read as follows: Data Plotting Cartography Mechanical CAD Electronic CAD Architectural CAD Business Graphics Flight and Driving Simulation Film Animation (2-D and 3-D) Video Animation (2-D and 3-D) Document Generation Video Games Display of Mathematical Objects Process Control Office Information Systems Free Hand Drawing and Painting Now we can decide why computer graphics has value in these applications. Reasons found or generally given include the following: Pictures could not be made otherwise Pictures can be made faster Pictures can be made cheaper by computer Pictures look different by computer Pictures look better by computer Internet computing e-commerce Internet based collaborative work and visualization C. The Graphic Process Models 5

6 Transformations Display Visibility Pixel driven rendering Ray-Tracing Voxel-Based Rendering Interactive & Collaborative Visualization A. Interactive Visualization B. Collaborative Visualization Conclusion and Future Trends Computer graphics visualization techniques for analysis have quickly become an active area of research and development. Beyond its most obvious aspects of the display of behavior, engineering analysis visualization involves issues such as interaction with a 3- D model, operations on result data and optimization of design variables. Overall, visualisation techniques have become part of a large trend in the engineering design field, leading towards a computing environment where design is an interactive process which encourages the exploration of design alternatives. 8.2 Optimization in Engineering Design and Optimisation Techniques Design is one of the principal activities of the practicing engineer. It refers to the development of an entity (an object, a system, a procedure, etc.) to perform a specified function. Among the goals of design is optimisation of the entity. To achieve that goal, we must have some way of deciding what makes one solution better than another. Such a measure or standard of comparison is known as a criterion. The process is further complicated by the existence of requirements that must be met if the design is to be acceptable. These requirements are the constraints of the problem Design Process 1. Define the problem. 2. Decide what output data are required to specify a solution to the problem. 3. Decide what input data are required to formulate the problem. 4. Develop models that can be used to obtain the required output from the input data. 5. Use the models to obtain a solution for a given set of input data. 6. Evaluate the solution in terms of the constraints and criteria. 7. Repeat as many parts of the process as necessary to obtain a satisfactory solution. 6

7 Case study: A problem description. Modeling the problem Optimization Methods Optimization of a design often involves steps 5, 6 and 7 of the design process. It typically requires us to determine a solution for a given set of input data, and to iterate to an optimum based on evaluations of previous solutions. We remark here that there are a variety of methods for the unconstrained global optimization problem. Some of them involve searches that require evaluation of only the criterion function. The more classical approach requires a solution of the system of equation that is obtained by setting the partial derivatives of the criterion function to zero. The root-solving methods would be useful with this approach. The presence of constraints rules out the direct zero-derivative approach because the optimum solution of the constrained problem does not generally coincide with a stationary point of the criterion function. Instead, we often find an optimum value with nonzero derivatives at a point on one or more of the constraint boundaries. Various Optimization Methods: Exhaustive searching Searching on constraint boundaries Lagrange multipliers Genetic algorithms Hamiltonian algorithms And other methods Numerical Methods for Optimisation Numerical Methods for Constrained Optimization - Any relationship that must be satisfied is a constraint. - Constraints are classified either as equality constraints or as inequality constraints. - Constraints limit the set of solutions from which an optimal solution is to be found. Numerical Methods for Unconstrained Optimization - Mainly for non-linear problems 7

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