Variational geometric modeling
with black box constraints
and DAGs
Paper by: Gilles Gouaty, Lincong Fang, Dominique Michelucci, Marc
Daniel, Jean-Philippe Pernot, Romain Raffin, Sandrine Lanquetin, Marc
Neveuc
A presentation by: Boris van Sosin
1
Introduction and motivation
Modeling with CAD systems is a multi-level process
Product
Component
Structure & surface details
Geometric primitives
Used by product designers
Used by CAD systems
2
Introduction and motivation
Parametric modeling:
• Vector of parameters: 𝑈 = 𝑢0 , 𝑢1 , … , 𝑢𝑛−1
• Shape: 𝐹 𝑈
• Designer modifies 𝐹 𝑈 by changing 𝑈.
3
Introduction and motivation
Variational modeling:
• Designer specifies shape function and constraints:
𝐹 𝑈
𝑐0 𝑈 = 0
𝑐1 𝑈 = 0
⋮
𝑐𝑚−1 𝑈 = 0
• Solver finds 𝑈 which satisfies constraints.
4
Introduction and motivation
Variational modeling:
• Often the design needs to minimize some criterion.
• E.g. cost, mass, physical stress, surface energy…
• Often, 𝑛 ≫ 𝑚.
•𝐹 𝑈
argmin𝑈 𝐺 𝑈
s.t.:
𝑐0 𝑈 = 0
𝑐1 𝑈 = 0
⋮
𝑐𝑚−1 𝑈 = 0
5
Introduction and motivation
Main tool:
• DAG representation
• Leaves are variables or constants
• Simple operators:
+
*
^
• +,-,*,/, power, root, etc.
• More Complex operators:
Var x
Var y
Const 2
• Bézier/B-spline control points, subdivision surface parameters, etc.
6
Related work
• Variational modeling introduced in: “Variational geometry in
computer-aided design”, V. C. Lin, D. C. Gossard, R. A. Light, SIGGRAPH '81
Proceedings of the 8th annual conference on Computer graphics and interactive techniques, 1981
• Feature based modeling introduced in: “Expert form feature
modelling shell”, J.J. Shah. M. T. Rogers, Computer-Aided Design, 1988
• “Modifying the shape of NURBS surfaces with geometric constraints”,
S.M. Hu, Y.F. Li, T. Ju, X. Zhu, Computer-Aided Design, 2001
7
Numerical Optimization
Numerically solving constraint problems / optimization problems
• Recall: single variable, scalar function 𝑓 𝑥 case.
• Observation: optimization problems are local extremum finding
problems.
• Can be solved as zero problems of 𝑓 ′ 𝑥 .
• Newton’s Method on 𝑓 ′ 𝑥 :
1
′
𝑥𝑛+1 = 𝑥𝑛 − 𝑓 𝑥𝑛 ⋅ ′′
𝑓
𝑥
8
Numerical Optimization
Numerically solving constraint problems / optimization problems
• Multivariate case: 𝑓 𝒙 is a scalar function of a vector variable 𝒙.
• Derivative ⇒ gradient vector
Inverse second derivative ⇒ inverse Hessian matrix
• Newton’s Method:
For finding extrema of 𝑓 𝒙 , find 𝒙 such that 𝛻𝑓 𝒙 = 0.
𝒙𝑛+1 = 𝒙𝑛 − 𝛾 𝑯𝑓 𝒙𝑛 −1 𝛻𝑓 𝒙𝑛
• where: 𝑯𝑓 𝒙𝑛
−1
is the inverse Hessian, and 𝛾 is a step-size constant.
9
Numerical Optimization
• Quasi-Newton methods are used when 𝑯𝑓 𝒙𝑛 is difficult to
compute.
• Hessian is approximated by matrix 𝑩𝑛 .
• Update step is 𝒙𝑛+1 = 𝒙𝑛 − 𝛾𝑩−1
𝑛 𝛻𝑓 𝒙𝑛
−1
−1
• 𝑩−1
𝑛 has an update step: 𝑩𝑛+1 = 𝑢𝑝𝑑𝑎𝑡𝑒 𝑩𝑛
• Examples:
• DFP (Davidon–Fletcher–Powell)
• BFGS (Broyden–Fletcher–Goldfarb–Shanno)
10
Numerical Optimization
• Pattern-search methods are used when even gradient is unavailable.
• A discretized gradient descent approach.
• Evaluate function at a finite set of points
in a pre-determined pattern.
• If the minimum/maximum is not at the center,
set new center to minimum/maximum.
• Otherwise, shrink pattern by half.
• “Optimization by Direct Search: New Perspectives on Some
Classical and Modern Methods”, T.G. Kolda, R.M. Lewis, V. Torczon,
SIAM Review, 2003
Image source: https://en.wikipedia.org/wiki/File:Direct_search_BROYDEN.gif
11
The DAG Representation
• Advantages:
• More general than simple Bézier/B-spline representation.
• Functions such as abs, sgn, min, max, 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.
Var x
Var y
Const 2
• Limitations:
• Closest point on curve/surface to a given point.
• The “if” (or ternary) operator.
12
From DAGs to Procedures
Solution: allow any procedure to be a “black-box” 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.
13
Solving With Black-Box Constraints
Problem formulation:
• Most generally:
• argmin𝑋 𝐺(𝑋)
s.t.
𝐹 𝑋 =0
is a vector of variables,
where 𝑋 = 𝑋0 , … , 𝑋𝑛−1
𝐺 𝑋 is a scalar function,
𝐹 𝑋 = 𝐹0 𝑋 , … , 𝐹𝑚−1 𝑋
is a vector function of constraints.
14
Solving With Black-Box Constraints
Problem formulation:
• Shortcuts:
argmin𝑋 𝐺(𝑋)
s.t.
𝐹 𝑋 =0
• If there is no 𝐹 𝑋 : unconstrained optimization ⇒ use BFGS.
• If only approximate solution is needed:
minimize 𝐻 𝑋 = 𝐺 𝑋 + 𝑤 𝑖 𝐹𝑖 𝑋 2 , for a very large constant 𝑤.
• If no 𝐺 𝑋 : minimize 𝐻 𝑋 = 𝑖 𝐹𝑖 𝑋 2 .
15
Solving With Black-Box Constraints
Problem formulation:
• argmin𝑋 𝐺(𝑋)
s.t.
𝐹 𝑋 =0
• Solved as Lagrange Multiplier problem: ℒ 𝑋, 𝐿 = 𝐺 𝑋 + 𝑖 𝐿𝑖 𝐹𝑖 𝑋
• Local minima of 𝐺 𝑋 , with constraints 𝐹 𝑋 are:
𝜕ℒ 𝑋, 𝐿
𝜕𝐺 𝑋
𝜕𝐹𝑖 𝑋
=
+
𝐿𝑖
= 0 ∀𝑘:0≤𝑘<𝑛
𝜕𝑋𝑘
𝜕𝑋𝑘
𝜕𝑋𝑘
𝑖
𝛻ℒ =
𝜕ℒ 𝑋, 𝐿
= 𝐹𝑖 𝑋 = 0
∀𝑖:0≤𝑖<𝑚
𝜕𝐿𝑖
16
Solving With Black-Box Constraints
• Constrained optimization:
argmin𝑋 𝐺(𝑋)
s.t.
𝐹 𝑋 =0
• Solved as Lagrange Multiplier problem: ℒ 𝑋, 𝐿 = 𝐺 𝑋 +
• Solved as unconstrained optimization:
argmin𝑋 H 𝑋, 𝐿
where: H 𝑋, 𝐿 = 𝛻ℒ 𝑋, 𝐿
2
=
𝜕ℒ 𝑋,𝐿
𝑘
𝜕𝑋𝑘
• Derivatives approximated with finite differences.
2
+
𝜕ℒ 𝑋,𝐿
𝑖
𝜕𝐿𝑖
𝑖 𝐿𝑖 𝐹𝑖
𝑋
2
17
Solving With Black-Box Constraints
Solving:
argmin𝑋 H 𝑋, 𝐿
where: H 𝑋, 𝐿 = 𝛻ℒ 𝑋, 𝐿
2
=
𝜕ℒ 𝑋,𝐿
𝑘
𝜕𝑋𝑘
2
+
𝜕ℒ 𝑋,𝐿
𝑖
𝜕𝐿𝑖
2
• Finds only local minima
• Result depends on initial guess.
• Design process assumptions:
• Product design is an iterative process.
• 𝑋 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.
18
Solving With Black-Box Constraints
Solving:
argmin𝑋 H 𝑋, 𝐿
where: H 𝑋, 𝐿 = 𝛻ℒ 𝑋, 𝐿
2
=
𝜕ℒ 𝑋,𝐿
𝑘
𝜕𝑋𝑘
2
+
𝜕ℒ 𝑋,𝐿
𝑖
𝜕𝐿𝑖
2
• No initial guess for 𝐿𝑖
• Solved by homotopy-inspired method.
• Continuous deformation while preserving topology.
• “Solving geometric constraints by homotopy”, Hervé Lamure, Dominique Michelucci, SMA '95 Proceedings
• Define a continuum of problems:
𝑃 𝑡 : argmin𝑋 𝐺 𝑋 = 𝑡𝐺 𝑋 + 1 − 𝑡 𝐺 0 𝑋
s.t. 𝐹 𝑡 𝑋 = 𝑡𝐹 𝑋 + 1 − 𝑡 𝐹 0 𝑋 = 0 for 𝑡 ∈ 0,1 .
19
Solving With Black-Box Constraints
Solving by homotopy-like method:
𝑃 𝑡 : argmin𝑋 𝐺 𝑋 = 𝑡𝐺 𝑋 + 1 − 𝑡 𝐺 0 𝑋
s.t. 𝐹 𝑡 𝑋 = 𝑡𝐹 𝑋 + 1 − 𝑡 𝐹 0 𝑋 = 0 for 𝑡 ∈ 0,1 .
• 𝐺 1 𝑋 = 𝐺 𝑋 ,𝐹
• Choose:
1
𝑋 = 𝐹 𝑋 . 𝑃(1) is the original problem.
𝑋 0 = 𝑋 = initial guess.
𝐹 0 𝑋 = 𝐹 𝑋 − 𝐹 𝑋 0 . Lagrange multipliers 𝐿 = 0 satisfy these constraints.
0
0 2
𝐺
𝑋 = 𝑋−𝑋
or 𝐺 0 𝑋 = 𝐶𝑜𝑛𝑠𝑡 ≠ 0 so that 𝑋 (0) minimizes 𝐺 0 .
Lagrange multiplier problem:
𝜕ℒ 𝑋,𝐿
𝑘
𝜕𝑋𝑘
2
+
𝜕ℒ 𝑋,𝐿
𝑖
𝜕𝐿𝑖
2
= 𝛻𝐺
0
2
+
𝑖
𝐿𝑖 𝛻𝐹𝑖
0
2
+
𝑖
𝐹𝑖
0
2
20
Solving With Black-Box Constraints
Solving by homotopy-like method:
𝑃 𝑡 : argmin𝑋 𝐺 𝑋 = 𝑡𝐺 𝑋 + 1 − 𝑡 𝐺 0 𝑋
s.t. 𝐹 𝑡 𝑋 = 𝑡𝐹 𝑋 + 1 − 𝑡 𝐹 0 𝑋 = 0 for 𝑡 ∈ 0,1 .
• 𝐺 1 𝑋 = 𝐺 𝑋 ,𝐹
• Choose:
𝑋
𝐹
𝐺
0
0
0
1
𝑋 = 𝐹 𝑋 . 𝑃(1) is the original problem.
= 𝑋 = initial guess.
𝑋 = 𝐹 𝑋 − 𝐹 𝑋 0 . Lagrange multipliers 𝐿 = 0 satisfy these constraints.
0 2
𝑋 = 𝑋−𝑋
or 𝐺 0 𝑋 = 𝐶𝑜𝑛𝑠𝑡 ≠ 0 so that 𝑋 (0) minimizes 𝐺 0
• Iterative step:
• Advance 𝑡 by a small increment.
21
• Formulate problem 𝑃 𝑡 and solve (e.g. with BFGS) with previous 𝑋 as initial guess.
Examples of Use
1. Surface passing through point.
• Initial surface: bi-quadratic
B-spline,
with 7x7 uniformly-spaced
control point,
and open-ended uniform KVs.
• First design iteration: constrain 𝑆 0.5,0.5 to pass through point 𝑝.
• Problem formulation:
𝐺 𝑋 = 𝐷𝑖𝑠𝑐𝑟𝑒𝑡𝑒𝐸𝑛𝑒𝑟𝑔𝑦 𝑆
𝐹0 𝑋 = 𝐺𝑒𝑡𝑋 𝐶𝑎𝑙𝑐𝑃𝑡 𝑆, 0.5,0.5 − 𝐺𝑒𝑡𝑋 𝑝 = 0
𝐹1 𝑋 = 𝐺𝑒𝑡𝑌 𝐶𝑎𝑙𝑐𝑃𝑡 𝑆, 0.5,0.5 − 𝐺𝑒𝑡𝑌 𝑝 = 0
𝐹2 𝑋 = 𝐺𝑒𝑡𝑍 𝐶𝑎𝑙𝑐𝑃𝑡 𝑆, 0.5,0.5 − 𝐺𝑒𝑡𝑍 𝑝 = 0
• Border control points assumed to be constant ⇒ 25 ∗ 3 = 75 variables.
22
Examples of Use
1. Surface passing through point.
• First design iteration: constrain 𝑆 0.5,0.5 to pass through point 𝑝.
• Problem formulation:
𝐺 𝑋 = 𝐷𝑖𝑠𝑐𝑟𝑒𝑡𝑒𝐸𝑛𝑒𝑟𝑔𝑦 𝑆
𝐺𝑒𝑡𝑋 𝐶𝑎𝑙𝑐𝑃𝑡 𝑆, 0.5,0.5 − 𝐺𝑒𝑡𝑋 𝑝 = 0
𝐺𝑒𝑡𝑌 𝐶𝑎𝑙𝑐𝑃𝑡 𝑆, 0.5,0.5 − 𝐺𝑒𝑡𝑌 𝑝 = 0
𝐺𝑒𝑡𝑍 𝐶𝑎𝑙𝑐𝑃𝑡 𝑆, 0.5,0.5 − 𝐺𝑒𝑡𝑍 𝑝 = 0
23
Examples of Use
1. Surface passing through point.
• First design iteration: constrain 𝑆 0.5,0.5 to pass through point 𝑝.
• Problem formulation:
𝐺 𝑋 = 𝐷𝑖𝑠𝑐𝑟𝑒𝑡𝑒𝐸𝑛𝑒𝑟𝑔𝑦 𝑆
𝐺𝑒𝑡𝑋 𝐶𝑎𝑙𝑐𝑃𝑡 𝑆, 0.5,0.5 − 𝐺𝑒𝑡𝑋 𝑝 = 0
𝐺𝑒𝑡𝑌 𝐶𝑎𝑙𝑐𝑃𝑡 𝑆, 0.5,0.5 − 𝐺𝑒𝑡𝑌 𝑝 = 0
𝐺𝑒𝑡𝑍 𝐶𝑎𝑙𝑐𝑃𝑡 𝑆, 0.5,0.5 − 𝐺𝑒𝑡𝑍 𝑝 = 0
• Result:
24
Examples of Use
1. Surface passing through point.
• Second design iteration: constrain 𝑆 𝑢, 𝑣 to pass through 𝑝.
• Problem formulation:
𝐺 𝑋 = 𝐷𝑖𝑠𝑐𝑟𝑒𝑡𝑒𝐸𝑛𝑒𝑟𝑔𝑦 𝑆
𝐹0 𝑋 = 𝐺𝑒𝑡𝑋 𝐶𝑎𝑙𝑐𝑃𝑡 𝑆, 𝑢, 𝑣 − 𝐺𝑒𝑡𝑋 𝑝 = 0
(ditto for 𝑌, 𝑍).
• Two new variables:
𝑢, 𝑣 ⇒ 25 ∗ 3 + 2 = 77 variables.
• Use previous design iteration as
initial guess.
25
Examples of Use
1. Surface passing through point.
• Second design iteration: constrain 𝑆 𝑢, 𝑣 to pass through 𝑝.
• Problem formulation:
𝐺 𝑋 = 𝐷𝑖𝑠𝑐𝑟𝑒𝑡𝑒𝐸𝑛𝑒𝑟𝑔𝑦 𝑆
𝐹0 𝑋 = 𝐺𝑒𝑡𝑋 𝐶𝑎𝑙𝑐𝑃𝑡 𝑆, 𝑢, 𝑣 − 𝐺𝑒𝑡𝑋 𝑝 = 0 (ditto for 𝑌, 𝑍)
• Two new variables: 𝑢, 𝑣 ⇒ 25 ∗ 3 + 2 = 77 variables.
• Use previous design iteration as initial guess.
• Result: discrete energy reduced by ~20%.
26
Examples of Use
2. Contact between two parametric surfaces:
• Two cylinder-like objects, each composed of six patches
• Four cylinder body patches + two top/bottom patches
• All patches are bi-quadratic B-spline surfaces
with 5x5 control points.
• G1 continuity is forced along cylinder edges and G0 between
cylinder and top/bottom.
27
Examples of Use
2. Contact between two parametric surfaces:
• Contact area between the two objects is assumed to
be inside a single cylinder patch.
• Represented as 17 discrete contact-point constraints in
parametric space.
• Constraint system:
𝐶𝑎𝑙𝑐𝑃𝑡 𝑆1 , 𝑢1 𝑖 , 𝑣1 𝑖 = 𝐶𝑎𝑙𝑐𝑃𝑡 𝑆2 , 𝑅𝑜𝑡 𝛼, 𝑢2 𝑖 , 𝑣2 𝑖
𝐶𝑎𝑙𝑐𝑁𝑜𝑟𝑚𝑎𝑙 𝑆1 , 𝑢1 𝑖 , 𝑣1 𝑖 = −𝐶𝑎𝑙𝑐𝑁𝑜𝑟𝑚𝑎𝑙 𝑆2 , 𝑅𝑜𝑡 𝛼, 𝑢2 𝑖 , 𝑣2 𝑖
Additional variables: control points, size and location
of contact circle…
Additional constraints: G0 and G1 continuity.
• Not listed explicitly in the paper.
28
Examples of Use
2. Contact between two parametric surfaces:
• Result:
• Discretization of contact
area can’t guarantee
non-intersection of
the surfaces.
29
Examples of Use
3. Contact between parametric surface 𝑆1 and subdivision surface 𝑆2 :
• Subdivision surfaces do not support 𝐶𝑎𝑙𝑐𝑃𝑡, 𝐶𝑎𝑙𝑐𝑁𝑜𝑟𝑚𝑎𝑙.
• Instead, contact is forced by:
𝐶𝑎𝑙𝑐𝑃𝑡 𝑆1 , 𝑢, 𝑣 = 𝐶𝑙𝑜𝑠𝑒𝑠𝑡𝑃𝑡 𝑆2 , 𝐶𝑎𝑙𝑐𝑃𝑡 𝑆1 , 𝑢, 𝑣
𝐶𝑎𝑙𝑐𝑁𝑜𝑟𝑚𝑎𝑙 𝑆1 , 𝑢, 𝑣 = −𝐶𝑙𝑜𝑠𝑒𝑠𝑡𝑁𝑜𝑟𝑚𝑎𝑙 𝑆2 , 𝐶𝑎𝑙𝑐𝑃𝑡 𝑆1 , 𝑢, 𝑣
• 𝐶𝑙𝑜𝑠𝑒𝑠𝑡𝑃𝑡 is very difficult to implement without
“black-box” nodes.
• Also requires G0 and G1 continuity constraints.
• 𝐶𝑙𝑜𝑠𝑒𝑠𝑡𝑃𝑡, 𝐶𝑙𝑜𝑠𝑒𝑠𝑡𝑁𝑜𝑟𝑚𝑎𝑙 could be used in
previous example.
30
Examples of Use
4. Deforming a plane to create a bump/hollow
• Start from a bi-quadratic B-spline plane 𝑆0
with 17x17 inner control point.
• Target surface 𝑆 must pass through a target
curve above the plane, and a target outline
curve.
• Outline curve is discretized into 30 points, and
target curve into 11 points.
31
Examples of Use
4. Deforming a plane to create a bump/hollow
• Constraints:
𝐶𝑎𝑙𝑐𝑃𝑡 𝑆, 𝑢𝑖 1 , 𝑣𝑖 1 = 𝐶𝑙𝑜𝑠𝑒𝑠𝑡𝑃𝑡 𝑆0, 𝑃𝑖𝑂𝑢𝑡𝑙𝑖𝑛𝑒
𝐶𝑎𝑙𝑐𝑁𝑜𝑟𝑚𝑎𝑙 𝑆, 𝑢𝑖 1 , 𝑣𝑖 1 = 𝐶𝑙𝑜𝑠𝑒𝑠𝑡𝑁𝑜𝑟𝑚𝑎𝑙 𝑆0, 𝑃𝑖𝑂𝑢𝑡𝑙𝑖𝑛𝑒
for outline points.
𝑇𝑎𝑟𝑔𝑒𝑡
𝐶𝑎𝑙𝑐𝑃𝑡 𝑆, 𝑢𝑖 2 , 𝑣𝑖 2 = 𝑃𝑖
for target points.
• Variables:
17 ∗ 17 ∗ 3 control point variables,
30 + 11 ∗ 2 = 82 UV coordinate variables (possible error in paper),
• Total: 213 constraints in 949 variables.
32
Examples of Use
4. Deforming a plane to create a bump/hollow
• Experimental results:
• Initially, outline curve constraints are satisfied, but
not 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 result
for initial draft.
33
Optimizations & 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: +,-,*,/,^,abs,min, max,…
Vector to/from scalar components…
Parametric curve/surface creation/evaluation.
Other: ClosestPt, ClosestNormal for parametric and subdivision surfaces,
if-then-else…
• 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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Optimizations & 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: +,-,*,/,^,abs,min, max,…
Vector to/from scalar components…
Parametric curve/surface creation/evaluation.
Other: ClosestPt, ClosestNormal for parametric and subdivision surfaces,
if-then-else…
• Tradeoff:
High-level operations can be compiled into low-level computer code.
Fine-grained operators are better optimized with partial evaluation.
35
Optimizations & Design Considerations
• Labels on “black box” nodes:
• Make them not entirely “black box”.
• 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 (“black box” nodes).
• Open problem.
• Is meaningful handling possible?
36
Optimizations & 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 “persistence” may deform, get disconnected or
disappear.
• 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 box…
• Open problem.
37
Thank you!
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