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Variational geometric modelingwith black box constraintsand DAGs

Paper by: Gilles Gouaty, Lincong Fang, Dominique Michelucci, Marc Daniel, Jean-Philippe Pernot, Romain Raffin, Sandrine Lanquetin, Marc NeveucA presentation by: Boris van Sosin

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Slide2Introduction and motivation

Modeling with CAD systems is a multi-level process

ProductComponentStructure & surface detailsGeometric primitives

Used by product designers

Used by CAD systems

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Slide3Introduction and motivation

Parametric modeling:Vector of parameters: Shape: Designer modifies by changing .

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Slide4Introduction and motivation

Variational modeling:Designer specifies shape function and constraints: Solver finds which satisfies constraints.

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Slide5Introduction and motivation

Variational modeling:Often the design needs to minimize some criterion.E.g. cost, mass, physical stress, surface energyâ¦Often, .

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s.t.:

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Slide6Introduction and motivation

Main tool:DAG representationLeaves are variables or constantsSimple operators:+,-,*,/, power, root, etc.More Complex operators:BÃ©zier/B-spline control points, subdivision surface parameters, etc.

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Var y

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Slide7Related work

Variational modeling introduced in: âVariational geometry in computer-aided designâ, V. C. Lin, D. C. Gossard, R. A. Light, SIGGRAPH '81 Proceedings of the 8th annual conference on Computer graphics and interactive techniques, 1981Feature based modeling introduced in: âExpert form feature modelling shellâ, J.J. Shah. M. T. Rogers, Computer-Aided Design, 1988âModifying the shape of NURBS surfaces with geometric constraintsâ, S.M. Hu, Y.F. Li, T. Ju, X. Zhu, Computer-Aided Design, 2001

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Slide8Numerical Optimization

Numerically solving constraint problems / optimization problemsRecall: single variable, scalar function case.Observation: optimization problems are local extremum finding problems.Can be solved as zero problems of .Newtonâs Method on :

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Slide9Numerical Optimization

Numerically solving constraint problems / optimization problemsMultivariate case: is a scalar function of a vector variable .Derivative gradient vectorInverse second derivative inverse Hessian matrixNewtonâs Method:For finding extrema of , find such that . where: is the inverse Hessian, and is a step-size constant.

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Slide10Numerical Optimization

Quasi-Newton methods are used when is difficult to compute.Hessian is approximated by matrix .Update step is has an update step: Examples:DFP (DavidonâFletcherâPowell)BFGS (BroydenâFletcherâGoldfarbâShanno)

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Slide11Numerical Optimization

Pattern-search methods are used when even gradient is unavailable.A discretized gradient descent approach.Evaluate function at a finite set of pointsin a pre-determined pattern.If the minimum/maximum is not at the center,set new center to minimum/maximum.Otherwise, shrink pattern by half.âOptimization by Direct Search: New Perspectives on SomeClassical and Modern Methodsâ, T.G. Kolda, R.M. Lewis, V. Torczon,SIAM Review, 2003

Image source:

https://en.wikipedia.org/wiki/File:Direct_search_BROYDEN.gif

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Slide12The DAG Representation

Advantages:More general than simple BÃ©zier/B-spline representation.Functions such as etc. are easy to represent.Efficient to re-evaluate if only some parameters are changed.Derivatives are computed by chain rule.Where they exist.Composition is trivial to compute.Disadvantages:Interval analysis.Limitations:Closest point on curve/surface to a given point.The âifâ (or ternary) operator.

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Slide13From DAGs to Procedures

Solution: allow any procedure to be a âblack-boxâ node in a DAG.Properties:More high level operators.Can still know on which variables a node depends.Composition is still trivial to compute.Efficient evaluation by compiling procedures.Disadvantages:No symbolic evaluation, only floating-point.No closed-form formula representation.No derivative computation.Can be approximated by finite differences.Definitely no interval analysis.

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Slide14Solving With Black-Box Constraints

Problem formulation:Most generally:s.t.where is a vector of variables, is a scalar function, is a vector function of constraints.

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Slide15Solving With Black-Box Constraints

Problem formulation:Shortcuts: s.t.If there is no : unconstrained optimization use BFGS.If only approximate solution is needed:minimize , for a very large constant .If no : minimize .

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Slide16Solving With Black-Box Constraints

Problem formulation:s.t. Solved as Lagrange Multiplier problem:Local minima of , with constraints are:

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Slide17Solving With Black-Box Constraints

Constrained optimization:s.t.Solved as Lagrange Multiplier problem:Solved as unconstrained optimization:where: Derivatives approximated with finite differences.

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Slide18Solving With Black-Box Constraints

Solving:where: Finds only local minimaResult depends on initial guess.Design process assumptions:Product design is an iterative process. from the previous design iteration used as initial guess.If no initial guess is provided, meta-heuristics are used:simulated annealing, genetic algorithms, swarm optimization, etc.

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Slide19Solving With Black-Box Constraints

Solving:where: No initial guess for Solved by homotopy-inspired method.Continuous deformation while preserving topology.âSolving geometric constraints by homotopyâ, HervÃ© Lamure, Dominique Michelucci, SMA '95 ProceedingsDefine a continuum of problems: s.t. for .

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Slide20Solving With Black-Box Constraints

Solving by homotopy-like method:s.t. for .. is the original problem.Choose: initial guess. . Lagrange multipliers satisfy these constraints. or so that minimizes .Lagrange multiplier problem:

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Slide21Solving With Black-Box Constraints

Solving by homotopy-like method:s.t. for .. is the original problem.Choose: initial guess. . Lagrange multipliers satisfy these constraints. or so that minimizes Iterative step:Advance by a small increment.Formulate problem and solve (e.g. with BFGS) with previous as initial guess.

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Slide22Examples of Use

1. Surface passing through point.Initial surface: bi-quadraticB-spline,with 7x7 uniformly-spacedcontrol point,and open-ended uniform KVs.First design iteration: constrain to pass through point .Problem formulation: Border control points assumed to be constant variables.

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Slide23Examples of Use

1. Surface passing through point.First design iteration: constrain to pass through point .Problem formulation:

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Slide24Examples of Use

1. Surface passing through point.First design iteration: constrain to pass through point .Problem formulation: Result:

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Slide25Examples of Use

1. Surface passing through point.

Second design iteration: constrain to pass through .Problem formulation: (ditto for ).Two new variables: variables.Use previous design iteration asinitial guess.

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Slide26Examples of Use

1. Surface passing through point.Second design iteration: constrain to pass through .Problem formulation: (ditto for )Two new variables: variables.Use previous design iteration as initial guess.Result: discrete energy reduced by ~20%.

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Slide27Examples of Use

2. Contact between two parametric surfaces:Two cylinder-like objects, each composed of six patchesFour cylinder body patches + two top/bottom patchesAll patches are bi-quadratic B-spline surfaceswith 5x5 control points.G1 continuity is forced along cylinder edges and G0 betweencylinder and top/bottom.

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Slide28Examples of Use

2. Contact between two parametric surfaces:

Contact area between the two objects is assumed tobe inside a single cylinder patch.Represented as 17 discrete contact-point constraints inparametric space.Constraint system:Additional variables: control points, size and locationof contact circleâ¦Additional constraints: G0 and G1 continuity.Not listed explicitly in the paper.

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Slide29Examples of Use

2. Contact between two parametric surfaces:Result:Discretization of contactarea canât guaranteenon-intersection ofthe surfaces.

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Slide30Examples of Use

3. Contact between parametric surface and subdivision surface :Subdivision surfaces do not support , .Instead, contact is forced by: is very difficult to implement withoutâblack-boxâ nodes.Also requires G0 and G1 continuity constraints., could be used inprevious example.

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Slide31Examples of Use

4. Deforming a plane to create a bump/hollowStart from a bi-quadratic B-spline plane with 17x17 inner control point.Target surface must pass through a targetcurve above the plane, and a target outlinecurve.Outline curve is discretized into 30 points, andtarget curve into 11 points.

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Slide32Examples of Use

4. Deforming a plane to create a bump/hollowConstraints: for outline points. for target points.Variables: control point variables, UV coordinate variables (possible error in paper),Total: 213 constraints in 949 variables.

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Slide33Examples of Use

4. Deforming a plane to create a bump/hollow

Experimental results:Initially, outline curve constraints are satisfied, butnot target curve constraints.In first iterations, target curve constraints are improved,while outline constraints are deteriorated.At about 50 iterations, all constraints tend towards zero.Stopped at 300 iterations.Considered by authors to be a good resultfor initial draft.

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Slide34Optimizations & Design Considerations

DAGs allow partial re-evaluation of expressions.Needs system for tracking and managing changes in variables.Timestamps for all nodes.Vocabulary of operators:Arithmetic: +,-,*,/,^,,, ,â¦Vector to/from scalar componentsâ¦Parametric curve/surface creation/evaluation.Other: ClosestPt, ClosestNormal for parametric and subdivision surfaces,if-then-elseâ¦Tradeoff:High-level operations can be compiled into low-level computer code.Fine-grained operators are better optimized with partial evaluation.

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Slide35Optimizations & Design Considerations

DAGs allow partial re-evaluation of expressions.Needs system for tracking and managing changes in variables.Timestamps for all nodes.Vocabulary of operators:Arithmetic: +,-,*,/,^,,, ,â¦Vector to/from scalar componentsâ¦Parametric curve/surface creation/evaluation.Other: ClosestPt, ClosestNormal for parametric and subdivision surfaces,if-then-elseâ¦Tradeoff:High-level operations can be compiled into low-level computer code.Fine-grained operators are better optimized with partial evaluation.

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Slide36Optimizations & Design Considerations

Labels on âblack boxâ nodes:Make them not entirely âblack boxâ.Label properties such as: continuity and differentiability, convexity, linearity, etc.How can the solver take advantage of such information?Open problem.Failures, infinite loops, timeouts:Can occur in procedures (âblack boxâ nodes).Open problem.Is meaningful handling possible?

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Slide37Optimizations & Design Considerations

Persistent naming:Designer may want to preserve features.Example: the depression in example 4.User may want to move/rotate depression, but not lose it.Feature without âpersistenceâ may deform, get disconnected ordisappear.Open problem.Interoperability:Results can be exported as mesh / spline / subdivision surface models.No simple operator history representation.Example: Cylinder -> apply deformation -> move to contact with a boxâ¦Open problem.

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Slide38Thank you!

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