Superconvergence and Error Estimation Superconvergence and Error Estimation


Finite Element Solutions


Fire-exposed Frame Problems

A thesis submitted for the degree of Doctor of Philosophy


James Alexander Kirby

Department of Mathematical Sciences, Brunel University

October 2000


When a fire reaches the point of flashover the hot gases inside the burning room ignite resulting in furnace-like conditions. Thereafter, the building frame experiences temperatures sufficient to compromise its structural integrity. Physical and mathematical models help to predict when this will happen. This thesis looks at both the thermal and structural aspects of modelling a frame exposed to a post-flashover fire.

The temperatures in the frame are calculated by solving a 2D heat conduction equation over the cross-section of each beam. The solution procedure uses the finite element method with automatic mesh generation/adaption based on the Delaunay triangulation process and the recovered heat flux.

With the Euler-Bernoulli assumption that the cross-section of a beam remains plane and perpendicular to the neutral line and that strains are small, an error estimator, based on the work of Bank and Weiser, has been derived for finite element solutions to small-deformation, thermoelastic and thermoplastic frame problems. The estimator has been shown to be consistent for all finite element solutions and asymptotically exact when the solution involves appropriate higher degree polynomials. The asymptotic exactness is shown to be closely related to superconvergence properties of the approximate solution in these cases. Specifically, with coupled bending and compression, it is necessary to use quadratics, instead of linears for the compression and twisting terms, to get a global O(h2) rate of convergence in the energy norm, some superconvergence properties and asymptotic exactness with the error estimator.


When a building is designed it must meet safety requirements that include provisions for fire protection. Although the building as a whole is considered the requirements apply to individual structural elements. The assumption is that if the individual elements are satisfactory then the whole building should perform at least as well.

The ultimate method of determining the performance of a structural element is the laboratory fire test as laid out in BS476 and ISO834. Such testing is expensive and time consuming. The designer must be pretty sure that the structure will pass the test to avoid the repeated costs. Hence the need for physical and mathematical models that can help to predict the outcome. Furthermore, assemblies of structural elements may be modelled that would be just too impractical to test in the laboratory. As computer power increases the structures that can be modelled become more complex. The ultimate goal must be an `all singing all dancing' computer program that simulates every aspect of a building's response to a real fire. This is not yet practical and we still rely on many mathematical idealogies that simplify the structural problem.

Always at the forefront of computational modelling has been the finite element method with its flexibility to cope with complex geometries and ease of application to any system of partial differential equations. Historically, engineers have led the way in finite elements, applying the method to a wide variety of thermal and structural problems. Meanwhile mathematicians have analysed the performance of the method and, more importantly, how to improve the results it provides. Chapter 3 of this thesis describes the finite element method and introduces some standard techniques in error analysis and error estimation.

The most important structural aspect of a building is its frame. The performance of the frame under the influence of fire exposure will dictate that of the building. The first simplification of the overall structural problem is to model the building structurally as its skeletal frame loaded with the weight of the walls and floors within. The frame is then modelled as an assembly of one-dimensional structures known as beams. The behaviour of each beam is governed by its cross-sectional properties, both geometric and physical (i.e. temperature and stiffness). It is this frame problem that is the focus of this thesis. The mathematical background was largely covered by Timoshenko in 1934 although practical applications were limited until the development of the computer later in the 20th century. Since then authors like Berg and Da Deppo, Nigam and Toridis and Khozeimeh have pioneered the work in computational frame analysis. The paper by Toridis and Khozeimeh in 1971 outlines a general finite element method for the elastic and plastic analysis of rigid frames under both static and dynamic loading. More recently, authors such as Terro and Wang have applied the finite element method to fire-exposed frame models. While the models have become quite sophisticated, allowing for large deformation and very realistic material models, these finite element solution procedures have not benefited from modern mathematical developments.

Essentially, the finite element method performs calculations on a discretization of the domain, called a mesh, which, for two-dimensional domains, is a tessellation of polygons, called elements. The method derives a piecewise polynomial that is continuous across the element sides and approximates the solution to the partial differential equations such that

|| e || C hp
where ||e|| is a measure of the error in the energy norm, h is a measure of the element size, p is the order of the polynomial and C is a constant, independent of h but inversely proportional to the smallest element angle. Since the error depends on h, a way of reducing the error is to reduce the size of the elements. As to where the mesh needs smaller elements, users of the finite element method have developed error indicators. These are numbers that are computable from the finite element solution and approximate || e || or some other measure of the error such that, globally,
C1 || e || h C2 || e ||
where h is the error indicator and C1 and C2 are constants. The calculation of h is performed on an element by element basis in such a way that
h2 = n

i = 1 
where n is the number of elements. In this relation the hi's represent each element's contribution to the global error and are used to determine elements that need refining. A well known error indicator uses the method of gradient averaging. It compares the gradient of the finite element solution with that of a smoother function obtained by interpolating the average gradient at the nodes. This smoother gradient function is an example of a recovered gradient. An error indicator using an alternative recovered gradient method is that of Zienkiewicz and Zhu. Other error indicators have been developed based on the difference between the applied forces and those calculated from the finite element solution; see, for example, Babuska and Rheinboldt and Bank and Weiser.

Download a PDF file of the complete thesis (1005k pdf file).