Scanning – A New Approach to Numerical Modelling of Structures
THE MYKSTUFOSS HYDROELECTRIC ARCH DAM
In Norway, more than 99% of the electricity is being produced by hydropower. This is achieved by a large number of water reservoirs with a vast number of dam structures. Due to the severe implications of failure, the design basis are strict, and the owners are obliged to monitor the structural conditions continuously. Reassessment of the load-carrying capacity and safety conditions of hydroelectric dam structures is frequently necessary. The purpose of the study is to demonstrate the capabilities of scanning with respect to efficient modelling for structural analysis.
Figure 2 Picture and digital model of the arch dam
CONCLUSIONS
Both structures, the King’s entrance and the Mykstufoss dam, were scanned during a three-days working period. The number of separate scans, 8 and 10, for the cathedral and the dam, respectively, were combined partly at the spot, and complete models where available the day after.
The models are currently being used as basis for FEM-modelling and structural analysis. This is a straight forward process, however data reduction is necessary, and this is an issue which needs to be addressed.
Scanning as basis for numerical modelling of existing structures will be a common approach, be it for reassessment purposes, for determination of load-carrying capacity of structures subjected to accidental events, or for structural documentation in general.
ACKNOWLEDGEMENTS
The scanning, modelling and visualisation was performed by the company DelfTech bv in the Netherlands. The skilful work of Willem van Spanje and Erwin Beernink is highly appreciated.
The demonstration studies were made possible thanks to the practical support and enthusiasm of Kristin Bjørlykke at NDR and Henrik Grosch in Statkraft Grøner.
REFERENCES
[1] K. V. Høiseth. Three-Dimensional Structural Scanning. SINTEF Report MT70 F01-252.
[2] Computer Graphics Perception (CGP) Software Manuals, Cyra Technologies Inc.
On the Q
2− P
1Stokes element
Daniele Boffi∗
Dipartimento di Matematica “F. Casorati”
Universit`a di Pavia, Pavia, Italy e–mail: boffi@dimat.unipv.it
Lucia Gastaldi Dipartimento di Matematica Universit`a di Brescia, Brescia, Italy
e–mail: gastaldi@ing.unibs.it
ABSTRACT
Summary In this talk we comment on the popular Q2− P1Stokes element. In particular, we recall that on meshes of general quadrilaterals two different definitions of the scheme are possible: a global pressure approach or a local one. The inf-sup condition for the former method is known to hold; we have proved that the same is true also for the latter one. However, the global approach is to be preferred because of a lack in the approximation properties for the local one. Numerical experiments confirms the theory.
We consider the standard mixed formulation for the Stokes problem find (u, p)∈ V × Q such that
a(u, v) + b(v, p) = F (v) ∀v ∈ V
b(u, q) = 0 ∀q ∈ Q,
where Ω is a polygon, V = H01(Ω)2, Q = div V , the bilinear form a(·, ·) : V × V → R is a weak form for the vector Laplace operator and b(·, ·) : V × Q → R is the Stokes bilinear form b(u, p) =−R
Ωp div u dx.
Given a sequence of quadrilateral meshes {Th} of Ω, we consider, as usual, the bilinear map FK from the reference square ˆK which defines a generic quadrilateral K ∈ Th. The widely used Q2−P1finite element method for approximating the Stokes problem consists in choosing Vh ⊂ V and Qh ⊂ Q as follows. Starting from the space of bilinear vectorfields on ˆK denoted by ˆV , we can define V (K) as the compositions of functions in ˆV with FK−1. Then the definition of Vhis the following
Vh={v ∈ V : v|K∈ V (K), ∀K ∈ Th}.
The pressure spaces Qhcan be defined in two different ways. Since no continuity is required, there is no need for composing reference functions with the inverse of FK; this observation gives rise to the so called global approach: functions in Q(g)h are “true” piecewise linears, namely
Q(g)h ={q ∈ Q : q|Kis affine, ∀K ∈ Th}.
The local approach consists in defining Qh in a similar way as it has been done for Vh. Let ˆQ be the space of affine functions on ˆK and denote by Q(K) the space of compositions of functions in Q with Fˆ K−1. Then Q(l)h is given by
Q(l)h ={q ∈ Q : q|K ∈ Q(K), ∀K ∈ Th}.
We make clear that Q(g)h = Q(l)h whenever FKis affine for any K ∈ Th(i.e., when the elements of Th are parallelograms); however, if K is a general quadrilateral (for which FKis not affine) then Q(K) contains functions which are not affine, since FK−1is not a polynomial.
The Q2− P1 Stokes approximation then reads
find (uh, ph)∈ Vh× Qhsuch that
a(uh, v) + b(v, ph) = F (v) ∀v ∈ Vh
b(uh, q) = 0 ∀q ∈ Qh,
(1)
with either Qh= Q(g)h or Qh = Q(l)h .
Both approaches have been widely used in the engineering literature. The global approach has been analyzed in [3, 4].
In [1] it has been shown that particular care has to be taken into account when dealing with quadri-lateral elements. It follows that the spaces Q(g)h and Q(l)h achive different approximation properties, namely
inf
qh∈Q(g)h
||q − qh||L2 = O(h2) inf
qh∈Q(l)h
||q − qh||L2 = O(h) (2)
for a smooth function q (the lack of approximation properties for the space Q(l)h cannot be avoided, even for q polynomial).
When Qh = Q(g)h it is known that problem (1) is stable. In [2] it has been proved that this is the case also when Qh = Q(l)h . Hence the standard quasi-optimal error estimate for mixed methods holds true
||u − uh||V +||p − ph||Q≤ C inf
v∈Vh, q∈Qh
||u − v||V +||p − q||Q. (3) Combining (3) with (2), together with standard approximation properties for the space Vh (for the space Vhoptimal second order approximation holds true) we get, for a smooth solution (u, p),
||u − uh||V +||p − ph||Q= O(h2) for the global approach,
||u − uh||V +||p − ph||Q= O(h) for the local approach.
Numerical results confirm the suboptimal rate of convergence for the local approach. Even though the suboptimality arises from the bad choice of the pressure space, the loss of one order of con-vergence involves also the computed velocities. This result is in good agreement with the general phylosophy of mixed methods.
REFERENCES
[1] D. N. Arnold, D. Boffi, and R. S. Falk. Approximation by quadrilateral finite elements. Math.
Comp., to appear. (1999)
[2] D. Boffi and L. Gastaldi. On the quadrilateral Q2 − P1 element for the Stokes problem.
Pubblicazione IAN-CNR, Pavia (Italy), 1194/00, (2000).
[3] V. Girault and P.-A. Raviart Finite element methods for Navier–Stokes equations. Springer–
Verlag, Berlin, (1986).
[4] R. Stenberg. Analysis of mixed finite element methods for the Stokes problem: a unified approach. Math. Comp., 42(165) 9–23, (1984).