Volker Berkhahn, Matthias Göbel Institute for Computer Science in Civil Engineering, University of Hanover, Germany

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1 Numerical Simulation of Hydrodynamics Based on B-Spline Surfaces Volker Berkhahn, Matthias Göbel Institute for Computer Science in Civil Engineering, University of Hanover, Germany Michael Piasecki Civil & Architectural Engineering, Drexel University, Philadelphia, (PA) USA Abstract: Many of the commonly used procedures to generate a mesh for numerical computations make direct use of depth measurement points, for example through a triangulation. Problems with these approaches include an uneven and undesirable distribution of computational nodes, which can result in too dense or too coarse mesh sizes. Additionally, they do not account for depth measurement errors, i.e. erroneous measurements are directly incorporated into the computation. This paper presents a method that overcomes these shortcomings, by first computing a free-from surface that models the bathymetry, from which any numerical mesh can then be generated. The method is based on b-spline surfaces that use so-called de Boor points for representation of the surface. It allows to filter out faulty depth measurements, and at the same time ensures a high degree of accuracy by placing more de Boor points in regions that feature a high degree of bed elevation change. Based on this free form surface a simple mesh generation method is realized: Finite element nodes are created on intersection points of iso-parametric surface curves. Since the parameters of these iso-parametric curves can be freely chosen, meshes with different node density could easily be created, irrespective of element shape, i.e. either triangular or quadrilateral. This method, together with a Petrov-Galerkin finite element method for solving the shallow water equations, is used to demonstrate the applicability of the approach using the Delaware Estuary as an example domain. In the present state of development the numerical results of the approximation and mesh generation process fulfil the essential requirements based on the hydrodynamic equations. 1 Introduction Commonly, the bathymetry of a computational domain is described by a set of measurement points that result from depth sounding surveys. These point sets include areas with high and low

2 point density depending on the measurement campaign. In addition, these point sets may include gaps and overlaps of measurement areas. As a result, simple triangulation of the existing data points may lead to unsatisfactory meshes that are either too dense or too coarse. Hence, in order to apply a simple mesh generator based on these point sets, a data reduction/generation has to be performed in order to generate an optimal mesh. Other element configurations, for example quadrilateral elements, need to first generate an appropriate mesh onto which the depth soundings are then projected. This is a time consuming effort and must be repeated each time a different mesh size is sought. In this article a different method of mesh generation is applied and the corresponding numerical results and the calculation efforts are compared. In this approach the measurement points of the bathymetry will be approximated by b-spline free form surfaces. To improve the approximation quality the locations of the control points will be fitted to the specific bathymetry geometry. Finally, a quadrilateral finite element mesh based on the iso-parametric curves will be generated. The new mesh generation procedure will be demonstrated on the Delaware Estuary. The estuary is approximately 9 km in length and features a considerable variation in cross-sectional depth and width distribution. Results are compared with monitoring station readings, in order to demonstrate the method's applicability. 2 B-Spline Surfaces The mathematical foundation of spline geometry modeling was developed in the early sixties (de Boor 1962). Due to the rapid progress in computer technology the methods of geometric modeling are well known in the area of engineering and computer science (Hoschek, Lasser 1992; Farin 1993). The b-spline surface b is expressed in terms of the global parameters u and v: N M b(,) uv = d N () un () v with N, M ; L, K E ij i= j= ij i j 3 N N ; d (1) The control points dij of the b-spline surface are called de Boor points. These control points characterize the geometric shape of the free form surface. The upper limits N and M of the two sums indicate the number of intervals between the de Boor points in the u and v parameter direction, respectively. In Equation (1), N i K (u) and N j L (v) represent the b-spline function as shape functions with respect of the parameters u and v. The upper indices K and L denote the degree of the b-spline functions. These b-spline functions are defined by recursive formulas. For the case of degree, the b-spline functions are given by Equatio n (2). 1. for u ui, u i+ 1 Ni () u = for i =,..., N+ K (2). else

3 Here u i and u i +1 denote the limits of the i th parameter segment. Only in this i th parameter segment the b-spline function N i (u) has a value greater than, i.e. a constant value of 1. The margins of all segments for the parameter u are collected in the knot vector u. This indicates the local influence of de Boor points on the shape of the corresponding b-spline curve or surface. Therefore the b-spline geometries are called segmented curves or surfaces. T u = u, un+ K+ 1 with ui ui+ 1 for i =,..., N + K (3) The b-spline functions N i r (u) of the degree r are defined with respect to the b-spline functions N i r-1 (u) of degree r-1. In addition, the recursive formula (4) shows that the elements u i of the knot vector u influence the shape functions. u u u u N () u = N ( u) + N () u for r = 1,..., N + K ; i =,..., N+ K r r i r 1 i++ r 1 r 1 i i i+ 1 ui+ r ui ui++ r 1 ui+ 1 (4) The b-spline function N i r (u) shows only a value that is greater than in r+1 of the parameter intervals. This ensures the local influence on the shape of the b-spline curve over r+1 intervals of any de Boor point. The b-spline functions N j (v) and N j r (v) are given in analogy to formulas (2) - (4). The definition of the b-spline surface (1) has to be an affine combination of de Boor points dij. This means, the sum of weighting factors of the control points has to be equal to 1. For a single b-spline function this requirement is satisfied within the interval defined by knots u K and u N+1. N K Ni ( u ) = 1. for u uk, u N + 1 (5) i= For the double sum over the b-spline functions with respect to the parameters u and v the valid parameter interval is defined by: N M Ni ( un ) j() v = 1. for u uk, un+ 1 ; v vl, vm+ 1 (6) i= J So, the b-spline surface given in formula (1) is restricted to the following parameter array. N M ij i j K N+ 1 L M+ 1 i= j= b(,) uv = d N () un () v for u u, u ; v v, v (7) The following properties of b-spline surfaces have a great advantage for the approximation of bathymetries: The segmentation of the b-spline functions implies the local influence property of the de Boor points. This permits to approximate large areas with good accuracy and with only one single b-spline surface.

4 The convex hull property assures that the b-spline surface lies inside the convex hull of the de Boor points. Even for large degrees of the b-spline functions oscillations, well known in the case of Lagrangean interpolation surfaces, could be avoided within the entire parameter interval. The diminishing variation property implies that no line crosses the b-spline surface more often than the corresponding de Boor point grid surface. This means, the b-spline surface is always smoother than the according control point grid. 3 Approximation of the Bathymetry To avoid problems with the topology of several free form surfaces the bathymetry of the investigation area will be approximated by only one single b-spline surface (Lichy, Berkhahn 1999). Therefore, even simple surface orientated mesh generators will produce suitable meshes. The equations for determination of de Boor points and the necessary assumptions can be derived in a straightforward fashion (Berkhahn, Lichy 2). In order to avoid an un-determined equation at least (N+1) (M+1) measurement points p have to be provided. Normally, however, a considerably higher number of measurement points are available. If not, the number of parameter intervals in u and v parameter direction has to be reduced. Uniform knot vectors u and v are chosen as follows: u= ui+ 1 ui = vj+ 1 vj = v for i=,..., N + K; j =,..., M + L (8) Related to local x and y coordinates the de Boor points d ij are chosen in a regular, equidistant grid. As a result, the local x and y coordinates are known, while the corresponding z coordinates are still unknown. In order to determine the z coordinates the measure points p are expressed as a point on the approximating surface b(u,v): p = b ( u, v ) (9) m m m This formula leads to the following equations for the x and y coordinates where p xm, p ym, d xij and d yij are known values: N M xm= x m m = xij i m j m i= j= p b ( u, v ) d N ( u ) N ( v ) N M ym= y m m = yij i m j m i= j= p b ( u, v ) d N ( u ) N ( v ) (1) Only the global parameters u m and v m of the measurement p m related to the b-spline surface are unknown. This leads to two equations for the determination of the unknown parameters u m and v m of every measurement point p m. These two equations for the parameters u m and v m are independent to the corresponding equations for the other measurement points. But it is impossible to transform the equations (1) in explicit form for unknown parameters u m and v m. So, these equations are solved iteratively.

5 After the determination of these parameters u m and v m for all measurement points the formula for z coordinate can be formulated in analogy to (1). N M zm z m m zij i m j m i= j= p = b ( u, v ) = d N ( u ) N ( v ) (11) Because every measurement point is influenced by (K+1)(L+1) de Boor points, equation (11) can not be solve explicitly for the unknown z coordinates d zij. This leads to an over-determined system of equations which is solved via the Householder transformation. 4 Mesh Generation Based on B -Spline Surfaces The bathymetry area under investigation is approximated by a single b-spline surface. For this free form surface, shape functions of degree N=M=3 in every surface segment are chosen. In order to improve the approximation quality, the number of segment in u and v parameter direction is increased iteratively. This iteration is continued until a pre-specified minimum allowable deviation between approximating b-spline surface and measurement points is reached. In land regions complementary points are added to the set of measurement points to avoid numerical problems in the approximation process. The nodes of the quadrilateral finite elements are created on the iso-parametric lines of the u and v parameters. This leads to elements outside of the estuary region. As a result, these surplus elements are deleted. To achieve a smooth boundary polygon the transitional elements between water and land are cut and the remaining parts of the elements are re-adjusted. The presented mesh generation procedure is applied to a part of the Delaware Estuary shown in figure 1. A sub-region of the estuary is shown in more detail in figure 2. Figure 1: Geometry of the Delaware Estuary ( measurement points) Figure 2: Shaded detail of the Delaware Estuary

6 The approximating b-spline surface, the measurement points, the control point grid and the corresponding quadrilateral finite element mesh are shown in figures 3-6. The geometry part of the Delaware Estuary consists of 197 measurement points. In contrast to this, the approximating b-spline surface is defined by only 26 * 26 = 676 de Boor points. This leads to a considerable data reduction. Figure 3: Approximating b-spline surface Figure 4: Approximating b-spline surface and measurement points The finite elements in figures 5 and 6 amount to 125 * 125 = The density and the number of elements can easily be varied by changing the u and v parameters of iso-curves. Figure 5: De Boor point grid of the approximating b-spline surface and corresponding finite element mesh Figure 6: Measurement points and finite element mesh 5 Hydrodynamic Model Shallow water or nearly horizontal flow conditions are applicable for a large number of rivers and estuaries in order to accurately describe the governing physics (Cunge, Holly, Verwey 1981; Fisher, List, Koh, Imberger, Brooks 1979). Conservation of mass and momentum under these conditions lead to the following system of equations (Katopodes 1984) in which U U U + A + A + F= (12) t t t

7 h Z 2 p p q p ; c u 2u ; uv v u ; + U= gh gn A= B = F = + (13) x q uv v u c u 2v h Z 2 q p + q gh + gn y 3 7 h where u, v are the velocity components and p and q are the volumetric flow rates in the x and y directions, respectively; g is the acceleration constant, n is the Manning roughness coefficient, and h is flow depth. The finite element solution can be expressed as { } U= N U (14) in which N is a 3*12 matrix whose elements are the bilinear shape functions N, interpolating the nodal values {U} of the solution vector, i.e. 1 Ni = ( 1 + ξξi) (1 + ηηi) ; i= 1,2,3,4 (15) 4 in which ξ and η are iso-parametric coordinates in the traditional finite element procedure. The adopted Petrov-Galerkin solution, in contrast to the traditional Galerkin formulation, however, does not take the weighting function identically to the shape function. Instead, a discontinuous test function is chosen, which appears as: N x The corresponding variational statement then reads N y T T N* = N+ εxa + εya (16) ne Ω T U U U (17) N* + A + A + F dω= t t t where n e is the number of elements, and Ω is the area of a single element. In Equation (16) the terms associated with ε x and ε y can be interpreted as a dissipative interface, whose strength is controlled by the magnitude of ε x and ε y. It has been suggested (Katopodes 1984) to use the following expressions:

8 ε x x x = ; εy = ( u+ v)15 ( v+ v) x x y y x= 2 + ; y= 2 + ξ η ξ η (18) The advantage of above expression is that artificial dissipation is only introduced locally, based upon grid size and wave speed. In this fashion, the dissipation level is allowed to vary according to the local characteristics of both the flow and computational conditions. As a result, the procedure is highly accurate with respect to numerical dissipation and dispersion. Time marching accuracy is achieved with a second order accurate Crank-Nicholson scheme, while the required iteration during each time step is carried out by a Newton-Raphson root-find. The finite element method described above, has been used in a number of applications, for example (Katopodes 1987, Piasecki 1998), and shown to be an excellent tool for the modeling of hydrodynamics in rivers and estuaries. The theoretical derivation of the hydrodynamical equations implies a finite element mesh with quadrilaterals. In addition, the simulation results will be improved if the elemente edges in direction of water flow are larger than the edges perpendicular to the main flow direction. It has to be verified, if these requirements from the hydrodynamic viewpoint are fulfilled by the presented method of mesh generation. 6 Numerical Results In this chapter numerical results of an approximating process of concrete measurement data of the entire Delaware River will be discussed. The nodes of the finite element mesh are generated on the iso-parametric curves of the u and v parameters of the b-spline surface. Because of quadrilateral parameter areas it is very easy to generate a regular quadrilateral finite mesh. But this quadrilateral parameter area leads to the serious disadvantage of a quadrilateral approximation surface. This quadrilateral free-form surface causes a quadrilateral boundary of the corresponding finite element mesh as shown in figures 5 and 6. This disadvantage is redressed by a cutting algorithm: at a user defined cutting level the finite elements are trimmed. This cutting algorithm guarantees quadrilateral elements even at the mesh boundary. For the examples presented in this paper a cutting surface at the level of +1. m is chosen. A small bathymetry area is approximated with acceptable effort of calculation time and memory usage: The over-determined system of equations (197 rows*676 columns for the example presented in figures 3-6) can solved via the Householder transformation easily with a standard personal computer. If the same approximation and mesh quality should be achieved for the entire Delaware River shown in figure 7 a system of equations with about 5 rows and 1 columns would have to be solved. This would lead to dramatic rise at computation effort, which cannot be covered by a standard personal computer. The partition of the bathymetry into smaller

9 sub-areas leads to a substantial reduction in computing effort. The assembly of the resulting submeshes leads to a total mesh with C continuity. In order to improve the approximation quality and to achieve elements adjusted to the main flow directions adaptive de Boor point grids are necessary. In the present state of development an algorithm for adaptive control points is implemented but the resulting free-form surface does not fulfil all of the requests. So, at the moment the de Boor grid still has to be adapted manually by the user, which requires a little experience. The geometric approximation of the Delaware River is based on nine b-spline sub-surfaces. In each sub-surface in u- and v-direction the degree of K=L=2 and the number of intervals N=M=24 are chosen. This leads to 5625 de Boor points for the approximation of over 5 measurement points. The finite element mesh shown in figure 7 consists of more than 16 nodes. Figure 7: Finite element mesh of the Delaware River In figure 8 a three dimensional view of the finite element view is shown. Characteristic for the Delaware River are a couple of small creeks, which should be represented with great accuracy. Problems with the approximation of the measurement points are evident in figure 8. If the cross section of a small creek consists of only one or very few measurement points, a comparatively wide mesh cross section is generated and the finite element mesh might not follow the intuitive course of the creek. Figure 8: Finite element mesh and measurement points

10 7 Conclusion The presented mesh generation method is based on measurement point approximations with b- spline surfaces. This is an easy and fast method to generate finite elements suitable for numerical simulations in hydrodynamics. Even in case of difficult bathymetries finite element meshes are generated, which fulfil the essential requirements based on the hydrodynamic equations. The future development will be focused on algorithms to generate adaptive de Boor grids and an easier handling of the sub-surfaces. Acknowledgments Finally, the authors thank Mr. Frank Sellerhoff (University of Hanover) for visualization of the Delaware Estuary (figures 1+2) and for his assistance in analyzing the measurement data. References Berkhahn, V., Lichy, C. 2. Bathymetry Modeling with Free-Form Surfaces Based on Adaptive de Boor Grids. Proceedings (CD-ROM) of Hydroinformatics 2 in Cedar Rapids (Iowa) USA. Cunge, J.A., Holly, F.M., Verwey, A Practical Aspects of Computational River Hydraulics. Pitman Publishing, London. De Boor, C Bicubic Spline Interpolation. Journal of Mathematics and Physics, 41, pp Farin, G Curves and Surfaces for Computer Aided Geometric Design: A Practical Guide. 3 rd Ed. Academic Press, London. Fisher, H.B., List, E.J., Koh, R.C., Imberger, N.H., Brooks, N.H Mixing in Inland and Coastal Waters. Academic Press. Hoschek, J., Lasser, D Grundlagen der geometrischen Datenverarbeitung. Teubner Verlag, Stuttgart. Katopodes, N.D Two-Dimensional Surges and Shocks in Open Channels. ASCE Journal of Hydraulic Engineering, 11, No.6 June. Katopodes, N.D Finite Element Model for Hydrodynamics and Mass Transport in the Upper Potomac Estuary. Report to the Metropolitan Washington Council of Governments. Lichy, C., Berkhahn, V B-Spline Surface Modeling with Adaptive de Boor Grids in Hydroinformatics. ISESS 99 in Dunedin / New Zealand. In: Denzer,R., Swayne, D.A., Purvis, M., Schimak, G. 2. Environmental Software Systems, Environmental Information an Decision Support. Kluwer Academic Publishers. Piasecki, M A Numerical Model for the Transport of Radionuclides Incorporating Cohesive / Non- Cohesive Sediments. Journal of Marine Environmental Engineering, 4, p