FINITE-ELEMENT VOLUMES By Thomas G. Davis j ABSTRACT: The finite-element-volumesmethod is a new earthwork volumestech-

nique quite unlike conventionalmethods. The algorithm provides automatic curvature and prismoidal correction using ordinarily available cross-section data in conjunctionwithhorizontalbaselinegeometry. The cross sectionsare approximated as a series of rectangular elements of equal, user-specifiedwidth. As this width approaches zero, cross-sectionalarea and centroid location approach that of the original cross section. Every element is assumed to transition linearly along an offset curve concentric with the baseline to an opposing element upstation or to terminate on a tapered offset curve when an opposingelement does not exist. The resultingvolumeelements are thus curvilinearwedges or frustumsof wedges. Linear, circular, and Cornu spiral baseline (clothoidalspline) componentsare accommodated by the method. Numericalexamplesshow excellentagreementwith exact results even when the mass componentsare not prismoidal. A general formula for the volumeof a curvilinearmass componentand a new, high-precision,prismoidal curvature-correctiontechnique are also presented. INTRODUCTION Traditionally, the problem of computing volumes associated with transportation alignments (roadways, railways, and waterways) has been accomplished by assuming that the required volume is given by the product of the length (difference in baseline stationing) and half the sum of transverse areas at the beginning and end of the mass component. This technique is called the average-end-area method and is, by far, the most widely used method of computing volumes. The average-end-area approach has the advantage that is easily understood and implemented. Unfortunately, the results are exact if, and only if, transverse area varies linearly with length and the baseline is straight. These conditions are seldom met in practice. The first issue, namely that transverse area does not, in general, vary linearly with length, may be overcome by collecting very closely spaced cross sections. W h e n this is done, the transverse areas at the beginning and end of the mass c o m p o n e n t are nearly the same and the assumption of linearity is made more realistic. The second issue, however, cannot be ameliorated by any a m o u n t of data collection. That is, if the baseline exhibits significant horizontal curvature, the average-end-area formula cannot deliver accurate results for arbitrary components of volume. When curvature, is to be considered, a correction technique based on averaging the eccentricities, as well as the areas, of the beginning and ending cross sections is typically employed. The mass c o m p o n e n t is then modeled as an average volume of revolution. While curvature correction can be made to yield accurate results with dense cross section data, the process of identifying individual components of volume is difficult to automate. If higher accuracy is required from sparse cross-section data, prismoidal correction may be used. The prismoidal-correction technique assumes that the volume components may be modeled as prismoids. The general prismoid 1Appl. Mathematician, CLM/Systems, Inc., 5601 Mariner Dr., Tampa, FL 33609. Note. Discussion open until January 1, 1995. To extend the closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on January 13, 1994. This paper is part of the Journal of Surveying Engineering, Vol. 120, No. 3, August, 1994. 9 ISSN 0733-9453/94/0003-0094/$2.00 + $.25 per page. Paper No. 7071. 94

is a solid figure such that transverse area varies cubically with length. This technique is used to greatest advantage in regions of cut-to-fill transition where average-end-area results are notoriously poor (Davis et al. 1981; Easa 1991; Moffitt and Bouchard 1987). The use of pyramid and frustum-ofpyramid formulas in transition regions is an example of prismoidal modeling (Easa 1991). Again, the identification of individual volume components is difficult to implement in computer code. Another existing volumetric technique is that of computing the volume between two digital terrain models (DTMs). While DTM differencing does not suffer from errors associated with prismoidal correction, the design DTM must be fairly dense in regions of horizontal curvature if accurate results are to be obtained. Moreover, relatively sparse cross-section data are often the only source for DTM creation. This typically leads to poor DTMs and correspondingly poor volumetric results. Even when dense, photogrammetrically derived data are available, the proper boundary geometry and triangulation can usually only be achieved by interactive editing of surface discontinuity strings or break lines. Bear in mind also that this paper focuses on transportation alignments; the cross section, for better or worse, is usually mandated as the basis for data collection, design, construction and payment. In the following sections, a new method of computing alignment volumes will be presented. The finite-element-volumes method is an algorithm that provides automatic curvature and prismoidal correction using ordinarily available cross-section data in conjunction with horizontal baseline geometry. The technique takes its name from the fact that volumes are computed by the summation of finite, three-dimensional elements. The finite-elementvolumes method is an exact solution of an approximate problem, while classical techniques are more often approximate solutions of exact problems. The method is a numerical integration technique and the result, a Riemann sum.

The new theory is developed by first considering the geometric model in a general fashion, without regard to the specific curvilinear components of the baseline. Next, the baseline-dependent element geometry is presented. Numerical examples introduce a general formula for the volume of a curvilinear mass component and a new, high-precision, curvature-correction method with which to compare finite-element-volume results. The errors associated with curvature correction and prismoidal correction are well documented (Davis et al. 1981; Easa 1991; Hickerson 1964; Kahmen and Faig 1988; Moffitt and Bouchard 1987). The numerical examples given here illustrate extreme conditions and demonstrate the proposed method's ability to return very accurate results with very sparse data. While curvature and prismoidal correction are not generally available in commercial software, finite-element volumes accomplishes both, with fewer limitations and less data. MODEL

Fig. 1 illustrates an orthogonal curvilinear coordinate [station, offset, elevation (STA, OFF, ELV)] system superimposed upon the more familiar Cartesian coordinate (X, Y, Z) system. Curvilinear coordinate systems are based on a planar curve called the baseline. Station values are determined as path lengths along this baseline from some fixed point of zero station. Offsets are measured in the plane of the baseline, the X - Y or STA-OFF plane, perpendicular to the forward tangent of the baseline. Facing in the forward direction of the baseline path, offsets to the left are signed negative, 95

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FIG. 1. OrthogonalCurvilinearCoordinateSystem while those to the right are signed positive. The offset depicted in Fig. 1 is negative. Elevations correspond directly to Z values. The S T A - E L V "plane" is that curved surface extruding above and below the baseline into the Z half-spaces. When reference is made to the STAELV plane of paths other than the baseline, it is meant that the named path serves as the baseline of yet another curvilinear coordinate system. In order to describe the geometry of masses located about the baseline, two planar curves are defined together with their associated profiles. Fig. 2(a) illustrates the planar, offset and tapered offset curves in the S T A - O F F plane. The offset curve is a constant, concentric offset curve with the baseline as directrix. The tapered offset curve is a transition curve between unequal offset values. The offset at any point on the taper varies linearly with respect to baseline length. The tapered offset curves serve to terminate the constant offset curves. Now consider "elevating" or assigning continuous elevation values to the offset curves as illustrated in Fig. 2(b). The resulting system of offset profiles, or longitudinal cross sections, define a surface in curvilinear coordinates. The superposition or intersection of two such surfaces defines a mass or volume. We will refer to one of these surfaces as the terrain surface and the other as the design surface. It is the volume, in cut and fill, between these surfaces that we wish to compute. While the baseline geometry is generally known completely and continuously as a horizontal alignment, the elevation data are usually collected as transverse cross sections in the O F F - E L V plane at discrete station values. In the finite-element-volumes model it is assumed that cross sections are normal (orthogonal) to the baseline and that cross sections exist at every nonlinear, taper-transition station. The baseline geometry together with these cross sections comprise the data. It is necessary to invent a rule by which elevation values transition along the longitudinal cross sections from one known value to another. In the finite-element-volumes model it is assumed that elevations transition linearly along the offset and tapered offset profiles as viewed in the S T A - E L V plane of the baseline. For the most part, this is consistent with assuming that the 96

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(b) FIG. 2. Offset and Tapered Offset Curves and Profiles: (a) Plan View; (b) Perspective View

baseline profiles (design and terrain) transition linearly from one known value to another. The exception is superelevation on a tapered offset. In this event, vertical curvature will arise even when the baseline profiles are linear. For this case, and when the baseline profiles are nonlinear, more cross sections need to be collected in order to adequately represent these vertical curves as a series of short chords. Implicit in the assumption of profile linearity is the existence of cross section data at every nonlinear, superelevation transition station.

Approximate Problem Fig. 3(a) represents a transverse cross section in cut and fill. Fig. 3(b) represents the same cross section after approximating the cut and fill areas as rectangles of uniform width. The height of each rectangle is given by the difference of design and terrain elevations at the midpoint. These rectangles are the transverse cross sections of the finite elements we will sum to obtain volumes. Clearly, as the element width decreases, the discrete, planar model agrees 97

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(b) FIG. 3. Continuous and Discrete Transverse Cross Section: (a) Continuous Cross Section; (b) Discrete Cross Section

more and more closely with the continuous model. Indeed, the discrete model converges rapidly with respect to cross-sectional area since the errors tend to be compensatory. In the current implementation of the finite-element-volumes algorithm, the maximum element width is user-specified. Internally, an element width is chosen such that the region between transverse cross-section pairs is spanned from left to right (Fig. 4). Exact Solution of Approximate Problem The volumetric problem is now one of computing and summing the individual element volumes. Every element is assumed to transition linearly (in the STA-ELV plane of the baseline) to an opposing element upstation or to terminate on a tapered offset curve when an opposing element does not exist. The resulting volume elements are thus curvilinear wedges or frustums of wedges. The length of an element is measured along the offset curve corresponding to the centroid of the element's transverse cross section. It is in the STA-ELV plane of this offset curve that the longitudinal crosssection area is computed. Element volumes are then calculated as the product of longitudinal area and uniform element width. For the approximate problem, this is an exact solution. This discrete, volumetric model is a Riemann sum of longitudinal elements. While computation-intensive, the model is very flexible and quite powerful when employed on a modern, high-speed computer. As in the OFF-ELV planes of the transverse cross sections, the discrete model converges to a continuous representation as the element width approaches zero. 98

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The number of elements required to produce volumetric convergence is higher than that needed for transverse area convergence but still easily attainable. A further simplification of the model may be achieved by considering Cavalieri's theorem in a plane: If two planar areas are included between a pair of parallel lines, and if the two segments cut off by the areas on any line parallel to the including lines are equal in length, then the two planar areas are equal (Eves 1991). Accordingly, it is not necessary to consider the terrain and design surfaces independently; only the difference in elevation between the surfaces, or signed height, need be known [Fig. 5(a)]. The foregoing assumption of independent linearity in the profiles may be weakened as follows. Along every offset and tapered offset line, the difference in elevation between the design and terrain surfaces varies linearly 99

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FIG.5. Cavalieri'sTheoremina Plane:(a)LinearProfiles;(b)NonlinearProfiles with respect to distance along the baseline. Thus, finite-element volumes will produce accurate results even when the profiles contain vertical curves, provided that the elevation difference is linear [Fig. 5(b)].

ELEMENTS Throughout the following it is assumed that the baseline is at least once continuously differentiable or "tangent." Continuous first derivative assures the existence of a curvilinear coordinate representation. Uniqueness of this representation is also an issue. Basically, cross-section lines must not cross one another, and curve offsets must be limited toward the center of curvature by the radius of curvature. Assuring both the existence and uniqueness of curvilinear coordinate representations for cross-section data will eliminate the possibility of gaps and gores in the model. When the baseline object is a line segment [Fig. 6(a)], the offset-line length is simply the baseline length, i.e. Lo = L 100

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When the baseline object is a circular curve segment [Fig. 6(b)], the offsetline length is given by Lo = L(1 + ek)

(2)

where e = eccentricity or plan distance from the baseline to the transverse area centroid; and k -- 1/r is the baseline curvature. The eccentricity is positive if the offset is away from the center of curvature and negative otherwise. Note that the sign convention for the eccentricity differs from the convention used to obtain curvilinear offset values. The eccentricity shown in Fig. 6(b) is positive. When the baseline object is a spiral curve segment [Fig. 6(c)], the offsetline length is given by t 0 = t

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where kl = 1/rl = initial baseline curvature; and k2 = 1/r2 is the terminal

101

baseline curvature. The eccentricity is positive if the offset is away from the evolute and negative otherwise. The eccentricity shown in Fig. 6(c) is positive. The line, circular curve, and spiral curve segment compose the geometric elements of a clothoidal spline (Walton and Meek 1990). If the line segment is considered as a degenerate Cornu spiral with zero curvature at either end, and the circular curve segment is considered as a degenerate spiral with the same curvature at either end, then (3) subsumes (1) and (2). If the transverse cross sections back and ahead have unequal minimum/ maximum offsets, some or all of the elements will be terminated on tapered offset curves (Fig. 7). Modified baseline stationing and initial and terminal element heights are computed by linear interpolation in the STA-ELV plane of the baseline. Any element, regardless of baseline geometry, appears as a trapezoid or pair of triangles in the STA-ELV plane of the baseline (Fig. 8). Whenever the initial and terminal heights differ in sign, the height curve passes through the datum. It is computationally convenient to consider this element as composed of two individual elements, one in cut and the other in fill. The

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102

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