PRODUCT and SYSTEM OPTIMISATION in ENGINEERING … · engineering analysis community it ... and require the use of Finite Element Analysis (FEA) ... inside concrete at a shear connection

FENET THEMATIC NETWORK

COMPETITIVE AND SUSTAINABLE GROWTH (GROWTH) PROGRAMME

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Industry Sector RTD Thematic Area Date Deliverable Nr

Product and System Optimization

8.10.2003 2

PRODUCT and SYSTEM OPTIMISATION in ENGINEERING SIMULATION

DELIVERABLE PART 2---PSO RTD

Professor Grant Steven

University of Durham [email protected]

Summary: A general overview of what the PSO group consider to be the scope of this technology and most of thecurrent methods available are described in this deliverable. The report is a more complete andreferenced version of the article that was published in FENET news in July 2003. This documentserves as a brief introduction to the complex field of optimisation, as such it only provide simpleindications to the non-expert practitioner on which method to consider.



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1 INTRODUCTION This paper has been written to present the authors view of what the title of the FENET thematic area means and what the various techniques are that compose the methods currently in use. The paper will therefore contain definitions and explanations that it is hoped will be useful to the FENET community. The obvious starting point is the title of the thematic area and the distinction between product and system. In the FENET and wider engineering analysis community it is understood that what is being considered are things that are technical in nature, and require the use of Finite Element Analysis (FEA) at some stage in their design, hence the FE in FENET. An engineering product is seen as an entity or part of an entity that results from the design process, such at the wing of an aircraft or an single rib of the wing or a single rivet fastener for the rib to the wing. This entity has to survive all that is asked of it for all of its life. Hence the application of the analysis process to make sure the product itself is right prior to production. In conjunction with the above definition of a product the word system is taken to mean the production/manufacture/processing that goes into making the product. This may also require extensive use of Finite Element Analysis as the resulting strength, stiffness, longevity of the product can be as much dependant upon the process(es) under which it is made, as upon the design itself. Also included in this definition are the processes that go into making the materials used in the product. An example would be a fabrication involving extensive welding, the steel being used has to be properly made, the welding preparations have to be correct, the welding has to be done correctly to ensure material continuity and the residual stresses resulting from the welding have to be known, (a very difficult task). Without this knowledge the performance of the product would be uncertain. The welding example could be regarded as a simpler example of a process, there are many even more elaborate ones, hydro-forming, deep drawing and so on. Most of these complex processes are Multi-Physics and their optimization goals are multi-criteria. Finally the word optimization appears in the title of this FENET thematic area, this is a word that means many things to many people. In the context of an engineering product or system I take it to mean that generally things are as good as they could possibly get for the specified environment(s) the product has to exist in. Whither it is perfect is another matter. The factors that could be taken into account to ensure perfection can be simply engineering ones, is it strong enough?, is it stiff enough?, will it last? or include manufacturing, financial, ergonomic, psychological issues. Perfection in energy conversion is zero loss, perfection in combustion is stochiometric, perfection in load transfer for a single environment is constant stress. For a real design with many environments and many constraints there is no such thing as perfection, but there can be for a design a perfect compromise, which however, is subjective to the analyst/designer. What the PSO thematic area is attempting to achieve is to investigate what methodologies are currently available for PSO. To look at the range of applicability for each method and to



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possible define what works best in which circumstances. It is also an endeavour to come up with some benchmarks that can be used by researchers, developers and users as a means of understanding the various qualities of the methods and the outcomes. 2 FORMS OF STRUCTURAL OPTIMISATION There are four distinct forms of structural optimisation. Each one has a different solution strategy. In a given design situation any combination of these forms can be present. 2.1 Topology optimisation. Topology optimisation exists where the actual form of the structure is unknown in advance. What is known are the environments the structure has to live in and the optimality criteria and design constraints that need to be applied. In Aerospace this could be the shape of the wing carry through box for a large civilian aircraft that also has under-carriage pick up points. There are several load cases for all the different flight regimes and several different sets of support conditions for in-flight and landing. The optimality criterions are most likely to be on stress and stiffness with possible subsidiary ones on frequency and buckling. In mechanical engineering, topology optimization could be to determine the topology of a support bracket for an under-bonnet car component. There are static and dynamics constraints as well as manufacturing ones to do with the costs of competing construction techniques, forging, stamping, welding, investment casting. In civil engineering, topology optimization can consist of the bi-material situation of steel bar inside concrete at a shear connection with the traditional question of where and how much steel to use. There are various static live and dead loads as well as some dynamic loading. The object of topology optimisation is to have no restriction on the final form of the structure. It is like starting a sculpture for a big block of material and chipping away till a topology emerges that best meets the criteria. The traditional single criterion has been the fully stressed design (FSD) where each part of the final structure is at the same stress. Whilst this seemed attractive, a little reflection will reveal that when there are multiple load cases and support cases there is no way a fully stressed design can be achieved. What is best is some form of weighted compromise, mathematically the weights turn out to be the Lagrange Multipliers associated with the degree of satisfaction of the individual criteria with the Kuhn-Tucker conditions applied to detect those inactive constraints at the optimum point Over the last few years methods have emerged that can detect an optimal topology under multiple criteria and multi constraints. The criteria consist of stress, stiffness, buckling load, frequency, moments of inertia. Indeed any physical quantity that can be measured together with any physical process that can be analysed can now be included in topology optimization There are currently three methods that are effective in solving commercial topology optimisation problems, and two more emerging research methods, these are all described below:



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Optimality Criteria. Here elements in a finite element mesh are altered or removed by applying optimality criteria derived by applying the duel theory of convex programming to a separable approximation of the design problem. References: :: Rozvany (1989). Homogenization. Here the micro-structural form of the material has different shapes governed by two or more local parameters. For each element in the FE mesh these parameters become the design variables and mathematical programming techniques with sequential quadratic programming used to determine the optimal material distribution. More recently the design variables have been simplified form micro-structural detail to imply the material density which is generally penalized in some way so that the result is a hard-kill situation. References :: Bendsoe (1989)(1995), Bendsoe and Kikutchi (1988), Maute and Ramm (1995). Evolutionary Structural Optimisation (ESO). Here a fine FE mesh is constructed and after subsequent analysis elements that are lowly stressed/not contributing/ineffective are slowly removed. The evolution rate is low and often many 100's of FE calculations are needed. Currently the range of element sensitivities covers every aspect of an elements performance and multi-criteria and multi-physics situations are resolved using Pareto type weightings. References :: Xie and Steven (1993)(1997), Hinton and Seinz(1995). Genetic Algorithms (GA=s). More recently these methods have had limited success in solving structural topology optimisation problems. Genetic algorithms are based on the theory of natural selection where the structural topology optimisation can be achieved by successive ranking of populations of elements and elimination the weakest. References Gage (1994), Kroo et al. (1994), Goldberg (1989), Osyczka (2002). Cellular Automata. This methods has been around for along time. It uses a building block approach with cells appearing when needed. The adaptation to structural mechanics has proved difficult since the base structure and all subsequent ones must be valid. In a sense it resembles the Bi-directional ESO method whereby material is added in the proximity of highly stressed material. Again the ground structure must always be valid. References :: Inou et al. (1994), Tatting and Gurdal (2000), Gurdal and Tatting (2000).

2.2 Shape optimisation Shape optimization comes in two distinct forms, one where there is some small region of detail than needs to be sculpted such that the maximum local stress is minimized; this is referred to as local shape optimization. Secondly the whole profile of a structure can be investigated to determine what is best; this is referred to as global shape optimization. With local shape optimisation the topology of the structure is known and there is some aspect of detail that is giving rise to a high stress, such as a fillet or a notch. The object of shape optimisation is to find the best shape that will have the best stress outcome. When there is only one load case then shape optimisation is equivalent to a Min-Max problem where the maximum stress on the surface of the region is minimised. In other words the shape of the local surface is altered till the maximum stress is as small as possible. Further reflection will demonstrate that this is equivalent to the surface becoming an equi-stress contour. When there are several load cases or support cases then this is no longer the case



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and the best that can be said is that the shape is the best compromise. References: Haftka and Grandhi (1985), Ding (1986), Falzon (1996)(1997). A typical situation where shape optimisation is present is the shapes of forks in trees, the shape of a fillet between two sections of different width. These are external surfaces. Indeed any of the geometries where the stress concentrations are present are suitable for consideration using shape optimisation. When surfaces are internal the shape optimisation process will find the best shape for the internal cut out, for example single or multiple holes in composites. A typical situation where global shape optimization is used could be when the product has to be make in a singly connected way. For example carbon fibre bike frames are best made from a single sheet rather than the conventional tubular construction. The optimization takes a sheet of carbon fibre as the design space, locates the loads and restraints and proceeds to nibble away at the sheet on the basis that the final shape is a mini-max single surface. An example of this is given in Xie and Steven (1997). As with topology optimisation there are several techniques available for solving the shape optimisation problem, these are listed below.

Mathematical programming. The shape is described mathematically by a series of parameters (design variables) and the objective function is described in terms of these variables. Very often the design variables are line vectors or control points from the surface being optimized and the strategy is to find the move distance for the vectors or points. The differentials of the objective with respect to the variables are obtained directly or by computation with a finite difference form for the differential. Second differentials are obtained for the Hessian matrix and conjugate gradient, steepest decent or quadratic programming search engine used to find the set of design variables that fit the design criteria. References: Pederson (1992), Kristensen et al. (1976). Evolutionary Structural Optimisation. The same basic algorithm as is used for the topology optimisation form of ESO is modified so that only surface element in the design region are available for removal, this is called nibbling. The effect of this is to undertake shape optimisation. The advantage is that there can be multiple load and constraint environments. References: Xie and Steven (1997). Simulated Biological Growth. This techniques uses a gambit whereby stress at the surface being shape optimised is converted to temperature and the surface allowed to move outwards, if stress high, or inwards if stress low. The motion is achieved by the thermal expansion equivalent of the fake temperature. Several iterations are needed. The method produces good results but, to date, only for a single load-case. References: Mattheck (1990).

2.3 Size Optimisation. The structure is defined by a series of sizes and dimensions. Combinations of these sizes and dimensions are sought that achieve the optimisation criteria. There are two major categories of problems in size optimisation.



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For Discrete structures. This includes pin and rigid jointed structures. This is the area that has received to most attention over the last forty years. A structure is defined and its loads and supports. The sizes of the members are adjusted according to the optimisation goal(s). Traditionally mathematical programming techniques have been used, recently OC techniques have proved successful, as has ESO. If member sizes can go to zero then they are unnecessary and a much reduced structure can be produced, this situation is sometimes called layout optimisation. Most of the historical work because it used mathematical programming techniques which need continuous variation in order to define gradients or sensitivities, is limited to cross sections of simple shapes. Where the whole choice of section is required as well as different ratios of major to minor axes geometric behaviours then heuristic methods such as GA’s have roved more useful. For Continuum structures. This form of optimisation includes, aircraft style structures such as carbon fibre laminates, stiffened panels, wing layouts, with spars and ribs where the structure can be describes as series of sizes or parameters, such at stiffener pitch, skin thickness, ply angle. Optimisation techniques are then used to find the combination of design variables that give the best result. Methods generally used are mathematical programming for continuous variations and genetic algorithms for discrete variations. ESO that treats whole components as a group has been successful for this type of structures where the initial design space is very highly overpopulated and these members least needed are evolved out. References: Lencus et al. (2001).

2.4 Topography optimisation This is the least studied form of structural optimization. In its simplest form it can be the drape of a shell surface in space that best meets the design criteria. For stadium roofing and other tent like structures this is an interesting area and generally mathematical programming techniques are the solution method. References: Ramm et al. (2000) 3 STRUCTURAL FORMS

There to be four major structural forms: 1D discrete structures, frames and the like; 2D continuum, 2D discrete and full 3D. The nature of the optimization problem for each form is described below.

Discrete structures. This includes pin and rigid jointed structures. The individual elements are often referred to as being 1D, because they can be located only by their two end points and only require cross sectional geometry and material values to characterize them. This is the area that has received to most attention over the last forty years. A structure is defined and its loads and supports. The sizes of the members are adjusted according to the optimisation goal(s). Traditionally mathematical programming techniques have been extensively used. More recently GA’s, ESO and other heuristic algorithms have been adopted successfully.



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If member sizes can go to zero then they are unnecessary and a much reduced structure can be produced, this situation is sometimes called layout optimisation. If the ground structure used for layout optimization is very dense then the optimization is essentially a topology one. 2D Continuum Structures. For this structural form the material locally is two dimensional as represented by a surface with thickness. It can lie in 3D space and thus be a shell. The thickness dimension is not part of the solution space. Analysis methods include, plane stress, plane strain, axi-symmetric, plates and shells. For this structural form there can be topology optimization where one can start from a big sheet of material and remove those parts that are not as important to the fitness function as the remaining material. There can be shape optimization where material can only be removed from the perimeter of the region and there can be thickness optimization where the initial region is the same but the thickness of the material changes and could disappear. Finally the position of the region in 3D space could be varied to find the stiffest shell geometry for a given ground projection. 2D Discrete Structures. This title is included to represent the large range of structures which are assembled for sheet 2D type material. Typical of this is the wing of an aircraft with lots of different skin thicknesses and potential locations for stiffening members and ribs. Another example is the fabrication of a railway bogey from different metal thicknesses and different external dimensions. The optimization problem here is to find the optimum set of the design variables that optimizes the fitness function. For such discrete problems traditional sensitivity mathematical programming techniques are still prove successful and genetic algorithms have also been found to be very adept when the solution response space is very messy. 3D Continuum Structures. Here the full laws of continuum mechanics apply and the structural optimization of such bodies is either a full topology optimization or the more restricted shape optimization. For such activities the homogenization method or some other element based sensitivity method, like penalized density or evolutionary structural optimisation 9ESO) have proved the most successful. All of the above. It is entirely possible that a real structure may consist of all of the four separate type above and still the optimization needs to be done whereby there may be topology optimization needed for some parts, size optimization in others and discrete structure in another. For this task the recent heuristic methods such as GA’s and ESO are suitable. The latter is particularly appropriate since it is element based both for sensitivity and for fitness.

4 DESIGN OPTIMIZATION

More recently a group of methodologies have emerged, some totally new some an adaptation of older methods, that I would categorize as Design and Process Optimization. They all rely on the incorporation of analysis engines inside a loop that explores the design space. In a sense they can be used to explore the design space. Behind each different method there are different reasons. The following few paragraphs will attempt to elucidate some of these.



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Robust Design Optimization. It is one thing to determine some optimal set of design variables, but another to be able to say that the design is robust. By robust is meant that a small variation in one design variable, or group of variables, will not cause a major shift in the response. In other words the minimum that represents the optimum is as flat as possible rather that a steep sided valley. Sensitivity analysis can be used here and graphical techniques such as clouds of performance points with the location of outlying points indicating extreme sensitivity. Also probabilistic or stochastic approaches can be incorporated to check that small possible variation in design variables, like the thickness of sheet metal will not cause major changes in the optimum performance of the design.

Reliability Based Design Optimization. There can often be a difference between the design for an object and how it is eventually made, material values can change, parts can be undersize or over, holes can be in different locations and many more. These methods can examine ranges of design data with probabilistic effects on occurrence and report on the most sensitive issues. The term Stochastic FEA is being used for this type of situation, in recent years the word fuzzy data has been used. Other techniques used here include Monte-Carlo Methods whereby data sets that span the range of potential variation are shot through the solver(s) and the resulting cloud of results again scanned for sensitivity or insensitivity. In addition the range of physical processed incorporated into the design can include, fluid, thermal, electro-magnetic as well as the traditional structural.

Design of Experiments (DOE) coupled with Response Surface Analysis (RSA). The DOE part of this is a traditional method first developed in Japan for manufacturing quality control. In essence it is a method that given the limits on the values of the design variables will select the minimum number of calls on the analysis engine(s) that when the response is charted there will be sufficient information to determine the optimal combination of the variables. It is based on the simple premise that if there is no great variation in any variable the response of the system to that, for all other variable in their own range, will be monotonic, will be convex or concave. Thus after the analysis engines have been called and the various objective functions can be plotted on multi-variable response surfaces to visualize the optimum points. In essence this is an optimized version of the traditional design space searches.

Genetic Algorithms. Recently these methods are being incorporated into general purpose design optimization environments. They compliment the DOE/RSA methods and the stochastic methods and have the advantage of being able to find global optimum regardless of starting data sets. They also can call upon a range of analysis engines and handle multi-criteria situations easily with any form of complex fitness functions.

5 SUGGESTED STRUCTURAL OPTIMIZATION SITUATIONS The table below is a Ashopping list@ of situations where structural optimisation can be applied in most structural contexts which are of interest to FENET. Most of the categories are described in a generic sense without specific applications in mind. At a later stage it is hoped to upgrade this document with a list of items to which some of the methods have been applied to the benefit of members of FENET.



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Optimisation Problem Class of optimisation, optimality criterion

Constraints Techniques Available

Varying plate thickness distribution at element level to minimise weight

Volume minimization, stiffness and/or strength maximization

Stress and/or displacement

MP ESO Homogenization

Fabricated structure with plates of varying thickness.

Strength and/or stiffness based

Buckling, stress and/or displacement

MP ESO

Varying plate thickness to maximise specific buckling load

Change of buckling load for change in element thickness sensitivity based

MP ESO

Shape of fillet/surface detail Stress based, mini-max problem

Fatigue Minimize maximum stress.

ESO Biological growth Mathematical programming

Shape of single or multiple cut-outs

Stress based, mini-max problem

Fatigue

ESO Biological growth OC

Varying plate topology for max or min frequency

Frequency shifting Volume, strength and/or stiffness, desired frequency

ESO

Spacing of stiffeners on aircraft style panels.

Stiffness and Buckling maximisation

Stress and/or displacement

Mathematical programming, PASCO, ASTROS, VIPASA(see Haftka ch13)

Lay-up of carbon composite panels

Stiffness based Strength based

Mathematical programming

Topology layout for minimum weight

Stress based ESO Homogenisation OC

Discovery of new topologies Multiple load case with stress based criterion.

ESO

Optimum form of shell in 3D space Stress based. Stiffness based

Sensitivity methods. Mathematical programming, ESO.

Optimum thickness on shell Stress based ESO Biological Growth Mathematical programming

Thermal stress minimisation due to Ct differences

Stress based ESO

Fibre placement in composites to optimise load paths

Stress Based Load path

Adhesive Joints Stress Based Shape ESO Mechanical joints Layout optimization Max load capacity GA’s Others from FENET members please respond

6. Concluding Remarks Because many of the optimization methods have an overlap of data and outcomes they can be found bundled into a common software environment. Such a bundling offers a way of



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exploring the design space with various tools and finding out which best suits the particular problem being studied.

The tasks of the PSO group within FENET is to::

• To characterize the various methods and their limits of applicability and scope

• To prepare benchmarks that can be used to test the efficacy of any method on a particular problem. This efficacy will measure both quality of result and the resources needed to find it.

• To determine the areas of the field that are not fully developed and to encourage research groups and/or the commercial vendors already developing the optimization tools to further extend the capabilities.

• To identify where there is expertise in the various forms of optimization and publish these for the benefit of the members.

• To prepare training material that can aid new and early users of optimization.

• To communicate the needs of analysts to software developers and vendors in this area.

7. REFERENCES

Bendsøe, M.P. (1995) Optimization of Structural Topology, Shape, and Material, Springer, Heidelberg.

Bendsøe, M.P. (1989) Optimial shape design as a material distribution problem, Struct.

Optim. 1, 193-202. Bendsoe, M.P. and Kikuchi, N. (1988) Generating optimal toplogies in structural design

using a homogenisation method, Comp. Meth. In Appl.Mech. Engng., 71, 197-224. Ding, Y. (1986) Shape optimisation of structures: a literature survey. Comput. Struct. 24(6),

985-1004. Falzon, B., Steven, G.P. and Xie, Y.M. (1996) Shape optimization of interior cutouts in

composite panels. Struct. Optim. 11, 43-49. Falzon, B., Steven, G.P. and Xie, Y.M. (1997) Mulltiple cutout optimization in composite

plates using evolutionary structural optimization. Struct. Eng. Mech., 5, 609-624. Gage, P. (1994) New approaches to Optimisation in Aerospace Conceptual Design, PhD

Stanford. (Also NASA CR-196695) Goldberg, D.E. (1989) Genetic Algorithms in Search, Optimisation and Machine Learning,

Addison-Wesley.



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Gürdal, Z., and Tatting B. T., “Cellular Automata for Truss Structures with Linear and Nonlinear Response," AIAA-2000-1580, Proceedings of the 41st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, April 3-6, 2000.

Haftka, R.T. and Grandhi, R.V. (1985) Structural shape optimisation: a survey. AIAA-85-

0772, AIAA/ASME/ASCE/AHS 26th Struct. Dyn. Mat. Conf., Florida. Hinton, E and Sienz, H (1995) Fully Stressed Topological Design of Structures using an

Evolutionary Procedure, Engineering Computations, 12, 229-244. Inou, N., Shimotai, N. and Uesugi, T. (1994) A Cellular Automaton Generating Topological

Structures, Proc 2nd European Conf on Smart Struct and Materials, Glasgow. Kristensen, E.S. and Madsen, N.F. (1972) On the optimum shape of fillets in plates subject

to multiple in-plane loading cases, Int. J. Num. Meth. Engng. 10, 1007-1019. Kroo, I, Altus, S., Gage. P., Braun, R. and Sobieski, I (1994) Multidiciplinary Optimization

Methods in Aerospace Vehicle Design , 5th AIAA/USAF/NASA/OAI Symp. On Multidiciplinary Analysis and Optimization, Panama City, Florida.

Lencus, A, Querin, OM, Steven, GP and Xie, YM. (2001) Aircraft Wing design Automation

with ESO and Group ESO, International Journal for Vehicle Design, 28(3), 98-111. Mattheck, C. and Burkhardt, S. (1990) A New Method of Structural Shape Optimisation

Based on Biological Growth, Int. J. Fatigue, 12(3), pp185-190. Maute, K. and Ramm, E. (1995) Adaptive topology optimization Struct. Optim. 10, 100-112. Osyczka, A. (2002) Evolutionary Algorithms for Single and Multicriteria Design

Optimization, Physika-Verlag, Berlin. Pedersen, P., Tobiesen, L. and Jensen, S.H. (1992) Shapes of Orthotropic Plates for

minimum Energy Concentration. Mech. Struct and Mach. 20(4), 499-514. Ramm, E., Kemmler, R., Schwarz, S.: Formfinding and optimization of shell structures.

Proc. of 'IASS-IACM 2000', Fourth International Colloquium on Computation of Shell and Spatial Structures, Chania, Crete, Greece, 4-7 June 2000.

Rozvany, G.I.N. (1989) Structural Design via Optimality Criteria, Kluwer Academic

Publishers, Dordrecht. Tatting, B., and Gürdal, Z., “Cellular Automata for Design of Two-Dimensional Continuum

Structures," AIAA-2000-4832, Proceedings of the 9th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, 2000.

Xie, Y.M. and Steven, G.P.(1993), A simple evolutionary procedure for structural

optimization, Computers and Structures, 49, 885-896. Xie, Y.M. and Steven, G.P.(1997), Evolutionary Structural Optimisation, Springer.

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PRODUCT and SYSTEM OPTIMISATION in ENGINEERING … · engineering analysis community it ... and require the use of Finite Element Analysis (FEA) ... inside concrete at a shear connection