The Bivariate Contouring Problem

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(1)The Bivariate Contouring Problem Thomas Grandine. . Bogdan Craciun. Quoc Thong Le Gia. . . Noel Heitmann. Miao-jung Ou . . Brian Ingalls. Yen-hsi Richard Tsai. . . July 28, 2000. Abstract An algorithm is presented for determining a connected component of the zero level set of a function

(2) , where

(3) is a bounded subset of . Two different numerical methods are employed and an error estimate procedure is indicated. Some examples that suggest possible applications are presented.. 1 Problem Description The problem of finding the zero level set of a function ! (with "$#&% ) frequently occurs in practice. The case "(')%+*&, has been extensively studied numerically (see [2] or [3]), while the study of other cases is still in an incipient phase. The present report is concerned with the case " '-%+*/. . There are several motivations for studying this problem, such as the problem of finding the intersection of hypersurfaces in !0 (which could be trajectories in time of evolving three dimensional surfaces) or generating a parametric representation of an implicitly defined manifold. Another possible motivation comes from the need to determine the surface envelope of a swept volume, which is defined in the following way: assume that we are given a surface that is moving or deforming in time 1. 32547698;:=<?> . A@ where 25BC is bounded D. the surface envelope is the boundary of the set EGF R. BC @ IHJKB7698;:=<?> with. F B. 1ML. 2C:=JNPO3Q. This work was done while the authors were participating in the Mathematical Modelling in Industry Workshop for Graduate Students at the Institute of Mathematics and its Applications at the University of Minnesota S The Boeing Company (thomas.a.grandine@boeing.com) T U California Institute of Technology (santa@ama.caltech.edu) V University of Pittsburgh (wallop@pitt.edu) W Rutgers University (ingalls@math.rutgers.edu) Texas A&M University (qlegia@math.tamu.edu) RXR SYS University of Delaware (miao@udel.edu) University of California, Los Angeles (ytsai@math.ucla.edu). 1.

(4) The Bivariate Contouring Problem. 2. Before describing the problem, it should be mentioned that several simplifying assumptions have to be made to ensure an attainable goal due to the time contraint of this workshop. For this reason, we assume solutions to other problems that would be contained in a fully realized algorithm, as will be noted in the following. These problems are mathematically interesting and will surely be the subject of future research. In mathematical terms, our problem can be set as follows. Given a smooth function is a bounded subset of ), we wish to parametrically represent the set (where EGF F LF B N ' 8O as the image of a function 698;: , > (for simplicity, we have chosen our parametric domain to be 6 8;: , > here). In addition to this, we need boundary conditions. Designing appropriate boundary conditions could be accomplished by an algorithm similar to the one described in [3] for the codimension equals one case. We will simply assume that we are given proper boundary conditions, i.e. the prescribed data lie on a connected subset of the zero level set of and that a consistent solution exists. Another difficult matter is uniqueness. Lack of uniqueness may result from two different sources. One is the possibility of several manifolds included in the zero level set of sharing the same (given) boundary. Such cases are often met in practice, as suggested by simple examples such as + 6 * .: .> @. . : . LF. LF. : : N '. . . L F , N . . *. . *. , N Q. We will assume that this is not the case (being avoidable by choice of appropriate boundary conditions). This and the assumption made in the previous paragraph ensure uniqueness of the manifold solution. F However, non-uniqueness of will automatically follow from the fact that generically this manifold has an infinite number of parametrizations. To single out a unique one, we need to impose additional conditions. Among many possibilities, we considered the following two: 1. isoparms traversed with constant velocity F . 2. area element is constant. '

(5) . L. F. N :. 4. PF . F . '

(6). L . N Q. (1). '

(7) Q. For the purpose of our numerical study of the problem, we chose to work with the first set of conditions. Although the second case is more natural geometrically, our choice is easier to implement and certainly mathematically sound. A direct consequence of (1) is F F. F F . '.

(8) . '.

(9) . . L L. N NQ.

(10) The Bivariate Contouring Problem. 3 . Differentiating the first equation with respect to and the second equation with respect to eliminates the unknown functions

(11) and

(12) and leaves us with F F. '. 8 '. 8Q. F F . . The result is a complete nonlinear partial differential algebraic system of % equations in % unknowns: LF . F F. N$'. 8 '. 8 '. 8:. F F . (2). with the solution made unique by the prescribed boundary data F L F L F L F L. 8: , :. :P8IN. '. : , N. '. N. ' N. '. . F. . F. . L. F

(13) L . L. B7698;: , >. . N. F L . . N. N N. . . B7698;: , >. . B7698;: , > B7698;: , > Q. Two numerical approaches for obtaining the solution to this boundary value problem were followed. The first used finite differences to approximate the partial differential equations in (2). The second used the finite element method to approximate the manifold, with collocation enforced at Gaussian points for the sake of faster convergence (as suggested by [1]). In both cases, Newton’s method was used to find a solution to the resulting nonlinear system of equations. In the following sections we will consider only the % ' case. The techniques used, however, work equally well in higher dimensions.. 2 The Finite Difference Approach This approach relies on approximating the partial derivatives in (2) in the interior of 6 8;: , > by finite differences. E We impose a uniform grid : O , A' 8;: Q Q Q : , : ' 8;: Q Q Q : , on the unit square, ! !"$#&% ' (" )% : lying on the boundary. We make use of the central difference and with : schemes for both the first and second partial derivatives: F L F. L. ' where + A. :. . . . :. . F L. %. N$' F L N$'. :. . . %. :. * *. F L. N *. . .,+ F L N *7.. ,.- (with analogous equations for. : . +. F . :. . and. *. *. N N . F . ).. F L . . :. . N.

(14) The Bivariate Contouring Problem. 4. The partial differential equations in (2) then become L F L L F L. % . :. : . *. * %. F L. N *. F L. N *. *. :. . * :. L F L F. 2. . . . . , . . LF. N$' @. and define the function . % . . %. . . . *. : . F *. . F. *. . @. %. . : . " . *. :. F L. for. LF. N$'. % . L F L. N=N. . for ' 8: Q Q Q : , : '&8: Q Q Q : F From now on, we will write 2. L F L. N=N. N *. .. . . . N. ". LF. 2 . . F. . . . . . . F. . . 2 . . . . . '. . * :. *. .. F. F. . . . . NN!'&8 NN!'&8. . . F. . . . . . . N NQ. . 2 .

(15) . . . N . . . . . L F. N. . . LF 2 N LF =N L F N 2 LF N 2 . .. . LF ". . . N. .. . LF. . .. . F " . . . . .. F. *. N L F N LF N. 2 . .. @ . F . . . *. :. . F L. N . = N. LF. . . . . LF. . . %. L F. 2 . F. . . . . : . F L. N . by . . % . LF. . F. F L. LF. N. . . . : . N . We introduce the difference operators. . .. F L. N *. 2. LF L F. 2. . ". N. N . ".

(16). N. . :. N. where the domain of the function is arranged to provide a nicely banded Jacobian. We use Newton’s method to solve the nonlinear equation ' " 8 . As known, this procedure F F is sensitive to a good initial estimate for the vector B @ at the grid points. In our F 1 examples, we used the outcome from Coons’ patch, which is the surface obtained by interpolating the given boundary data: 1 L :. . N. L ' *. ,?* L. . ,M*. N . F. . L. N ,?*. F. L. N F N. L. . L. N . 8IN *. L F. ,?*. . . L. N. F

(17) L L. , N*. N ,K*. Finally, we iteratively solve F . %. !'. F . *6 . LF . N >. . LF . N :. . F. N. F. . . L L. N , N *. L. ,?*. . N. F. . L. 8 NQ.

(18) The Bivariate Contouring Problem. 5. where is the Jacobian of . This Jacobian is a of about . Its structure is of the form . by . banded matrix with bandwidth. 0. 100. 200. 300. 400. 500. 600. 0. 100. 200. 300 400 nz = 4545. 500. 600. It is of interest to note that the band width of is dependent only on . This may allow a significant improvement of resolution for some problems without the addition of commonly expected computational expense.. 3 The Finite Element Approach The finite element approach attempts to compute the projection of the solution onto a chosen finite dimensional function space. We chose this space to be the one spanned by the tensor product of fourth order B-splines built on the sets of knots

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(21) and.

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(23) . ! " "# $ $%. . '& (& 32 415 ’s6have "62 415 the same meaning as in the previous section. where all ’s 1and 0 If )+*-,/.0 denotes the vector formed by all the coefficients of the solution in this function space, we have that " : . ; = ? < > @ > @ 789 &BA A ) &ED 5 @ "62 C D 52 83F & 9 G;3H C 9 ;I. C .00.

(24) The Bivariate Contouring Problem. 6. ' .( where and ' .( are the dimensions of the B-spline function spaces in the and directions, respectively. L L We force the equations in (2) to hold on the collocation points : N , G* .: , and are the roots of the quadratic Legendre polynomial, appropriately */. N : where % * % translated and scaled in the 6 : > and 6 : > intervals. This has proven successful in the one dimensional case, as mentioned in the first section. The functional equalities in (2), evaluated on these points, yield the following sets of nonlinear algebraic equations in :. . # .

(25). . and

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(27) . . ). #. . .

(28). ) . . @ ,. ) % . @ . %. )% . # @ . %. . . . . ) %. NN '. . . * . L . 8;:. . L . . (3). .:. L . N . . ,!. ) % . . L . . . @ . . N 2. ,. . . @ . L . *7.:. . . . . . . ) .

(29). ) % . ) . . . L. .

(30). .

(31). . # .

(32). . . #. ) .

(33). # . L

(34). . *. N#. ,!. L . . L . N2 .:. . N2 . *7.:. . . NA'/8;:. ,". L$ . N2 . ,%. %. L . N 2. L$ . . N ' . *7.. *. 8: .: (5). where '& ' & P: & : & N C B @ . L L @ To these * . N * . N equations we add the equations which result from the boundary conditions: # .

(35). . #. -.

(36). . . . #. .

(37). . ) % . @ . . . . . )% . . . . . . %)(. . . L . 8IN 2. . L . . N2. L. . L . ,!. N 2. F (

(38) L . N!'. 8;:. ,,+. ;:. F ( L *. . , :. *. , N *. .:. . 8;:. , ,+. ;:. L . . . N '. 8;:. , ,+. ;:. F ( L . *7.:. (6). N '. F ( . 8IN *. L. . , :. N * . ,. . *. L . , N2. . N * . 8*". %)(. . L . . ) % . L . 8*". %)(. ) @. . . . ) . . . @ . %)(. . ). .

(39). )% . @ . .

(40) . # .

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(42). ) .

(43). . N!'/8;:. ,,+ . ;:. (4). (7). (8). (9).

(44) The Bivariate Contouring Problem . . . 7. . #. ). ' ' 8 and ' '&8 . This is now a closed system of where equations with as many variables. As in the previous section, we attempt to solve this system with Newton’s method. For this, we first need to compute the derivatives of the left-hand side of the system with respect to & : . ' , Differentiation of the left hand side of (3), (4), (5), (6), (7), (8) and (9) with respect to ) % % respectively yields . @.. . L . L . N 2. . N=. # . . %. . . L . . . #. .

(45). . ) )% . @. . %. . . . for ( . and . (. . (. . . ( . . L. . . . L . . . . 8IN 2 . , N 2 L . L . . N 2. N2. L . L. L. . N . L . . L . ,. N . L . . . . L . L . L . N 2. *7.: , . N . . *. .: , . N# . L . . *. for. 8*. N. for. 8*. 8IN. for. ,GN. for. .: , ". L . N=N 2. L$ . 2 . ;:. *. , : , ,+. ;:. , . *. .: , ,+ . , . *. .: ,,+. ;Q. . J . N 2 . *7.: , . , : , ,+. . L$ . N2. .: , *. . . * . N=N. J . L L L$ N 2 N2 N . N. L . . L . , . for.

(46). ) %. . # @ * .: , ". . . ) % . @ . ) .

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(49). # . L

(50). L J . N . N=N . 4 Creating Boundary Conditions We have formulated a partial differential algebraic equation as a boundary value problem. In addition, two numerical methods coupled with Newton’s method have been presented. In order to generate the initial surface from Coons’ patch, we need to find the boundary curves. defines Let be the bounded domain of interest and implicitly as its zero E level set. The boundary curves may be found by finding the intersection of the ' 8O and E ' 8O3Q As mentioned in the earlier section, this can be formulated into a DAE and solved by the method described in [2]. . . . .

(51) The Bivariate Contouring Problem. 8. Another approach follows from the observation that the boundary curves are tangent to the normals of both surfaces. This condition can be written mathematically as F '. . . . 4. . 4. . Q. . We can either formulate the above equation as a initial value problem or a boundary value problem, and solve it with an appropriate ODE scheme.. 5 Error Estimation An appropriate criteria for error estimation is the distances from the points on the numerically obtained surface to the zero level set of . This can be achieved by first calculating the distance to the zero level set of . We do so by using the results in [4]. function C Consider the partial differential equation . . L. +% '. . . L N K , *. . N :. L. where +% N gives the sign of . Solving the above equation to steady state provides a function ' , , since convergence occurs when the right hand side is zero. with the property that The sign function controls the flow of information in the above; if is negative, information flows one way and if is positive, then information flows the other way. The net effect is to straighten out the level sets on either side of the zero level set and produce a function with ' , corresponding to the signed distance function. F L We measure the error of our approximation surface : N as . . . . . . . . . LF L ). . :. . 1 N=N :. 1. where is the surface element. Geometrically, the error measure the. exact surface and the approximation.. . 6 Examples We started with a simple example, for testing purposes. Consider . + A@. :. . LF. : :IN '. F. . *. , :. along with the following boundary data: F F. F. '. , : '. , : & ' 8: ' L '

(52) N: ' N: ' . . L L '

(53) N : ' N: ' . . L . F. '&8: '. . . for for . 8 ,. for. . . for. . B. 6 8;: , > B. 6 8;: , > B. 6 8;: , > B. 6 8;: , > Q. distant between the.

(54) The Bivariate Contouring Problem. 9 2. 2. f=x +y −1. 1 0.8 0.6 0.4 0.2 0 0 0.1 0.2 0.3 0.4. 0. 0.5. 0.2. 0.6 0.4. 0.7 0.6. 0.8 0.9. 0.8 1. 1. < The surface < solution to this problem is a quarter cylinder of radius one, between the planes and . By using code built on either the finite difference scheme or the finite element idea, we were able to find out approximations of the following parameterization of this surface:. 9 . 7 <. ;I. ).

(55) 9 <. G;I. <. for. . * . *. pictured in the figure above. We next attempted calculating more difficult examples, like: . ; < $ 7 9 9 7 7 7 ;) 9 , . , ). . . . . . ; ) '9. ;#. In just a few iterations, our code calculated the parameterization pictured in the figure below. In the initial running, we found damping was needed in the first few Newton iterations, in order for the solution to converge..

(56) The Bivariate Contouring Problem. 10. f = 1−2x2−50x(1−x)y(1−y)z(1−z). 1 0.8 0.6 0.4 0.2 0 0 0.1 0.2 0.3 0.4. 0.4. 0.5. 0.45 0.6. 0.5 0.55. 0.7 0.6. 0.8. 0.65. 0.9 1. 0.7 0.75. Finally, we present a practical application, that of computing the surface envelope of a swept surface, which is defined as the boundary of the set 7 ;I%. with 7 * 9 * , . * ; where 9 is our time evolving surface. A characterization for all the points situated on is that; the velocity should be perpendicular to the normal at the surface. Thus, if the surface 9 is given parametrically by and , the appropriate condition is 9

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(58) ;

(59) < : ; This can be interpreted as the zero level set of a real function defined on the 9 space and computed using either one of our codes. 9 : ; space of a surface envelope for a certain 9 ; Pictured below is the representation in the described by its spline coefficients. Using the results of our code, we were able to build up a three dimensional movie (see attached envelope.wrl file), a snapshot of which is shown on the last page.. .

(60) The Bivariate Contouring Problem. 11. 1 0.8 0.6 0.4 0.65. 0.2 0.6 0.55. 0 1. 0.5 0.8. 0.45 0.6. 0.4. 0.4. 0.35. 0.2 0. 0.3. References [1] de Boor C., Swartz B., Collocation at Gaussian points, SIAM J. Numer. Anal. 10 (1973), 582-606. [2] Grandine T. A., Applications of contouring, SIAM Review 42 (2000), 297-316. [3] Grandine T. A., Klein F. W., A new approach to the surface intersection problem, Comput. Aided Geom. Design 14 (1997), 111-134. [4] Sussman M., Smereka P., Osher S. J., A level set method for computing solutions to incompressible two-phase flow, J. Comput. Phys. 114 (1994), 146-159..

(61) The Bivariate Contouring Problem. 12.

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