We describe a method of constructing a new kind of splines with compact support on R. These basis functions consisting of a linear combination of the cardinal B-splines of mixed orders enable us to achieve simultaneously a good sampling approximation and an interpolation of any smooth function.

An accurate and efficient heterogenous three-dimensional computational model is developed for simulating the drying of wood. The complex macroscopic drying equations comprise a coupled and highly nonlinear system of partial differential equations. Due to the heterogeneous nature of wood, the physical model parameters strongly depend upon the local pore structure of the medium, the wood density variation within growth rings and the local variations in primary and secondary system variables. In order to provide a realistic representation of this behaviour, previously determined parameters derived using sophisticated image analysis methods and homogenisation techniques are used. Results are presented for a first generation virtual board description, where the board material properties vary along the section according to the pith position that defines the radial and tangential directions. These variations are assumed fixed throughout the longitudinal direction. The development of an accurate and efficient computational model requires the consideration of a number of significant numerical issues, including the virtual board description, the mesh design, the discretisation process, accurate flux approximations, and finally the solution of a large, nonlinear system. Each of these issues will be explored in this paper for the case of low temperature drying of softwood. Nomenclature c molar concentration (mol/m 3 ) D b bound liquid diffusivity (m 2 /s) D v effective vapour diffusivity (m 2 /s) g gravitational acceleration (m/s 2 ) h enthalpy (J/kg) h heat transfer coefficient (W/m 2 /K) h 0 vap latent heat of vaporisation (J/kg) J e energy flux (J/m 2 /s) J w liquid flux (kg/m 2 /s) K eff effective thermal conductivity (W/m/K) k m mass transfer coefficient (m/s) k w relative liquid permeability K w absolute liquid permeability (m 2 ) M v molecular weight of vapour (kg/mol) n normal vector (m) N number of mesh nodes P c capillary pressure (Pa) P w liquid pressure (Pa) r flux limiter sensor S w liquid saturation (m 3 /m 3 ) t time (s) T temperature (°C) V volume (m 3 ) v w liquid phase velocity (m/s) X moisture content (kg/kg) X average moisture content (kg/kg) X fsp fibre saturation point (kg/kg) x v molar fraction of vapour (mol/mol) Greek symbols Dh w differential heat of sorption (J/kg) dt discrete time step (s) dV CV control volume volume (m 3 ) e g gaseous volume fraction (m 3 /m 3 ) k w liquid mobility (m s/kg) l w liquid dynamic viscosity (kg/m/s) q density (kg/m 3 ) q 0 wood density (kg/m 3 ) r flux limiter function / porosity (m 3 /m 3 ) 382 S.L. Truscott, I.W. Turner / Applied Mathematical Modelling 29 (2005) 381-410

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In this work the control volume finite element (cvfe) method is used to discretise a two-dimensional non-linear transport model on an unstructured mesh. First and second order temporal weighting, combined with various flux limiting techniques (spatial weighting) are analysed in order to identify the most accurate and efficient numerical scheme. An inexact Newton method is used to resolve the underlying discrete non-linear system. In computing the Newton * School of Mathematics, Queensland University of Technology, Brisbane, Contents C760step the performance of the preconditioned iterative solvers bicgstab and gmres, in conjunction with a two-node Jacobian approximation is also examined. A linear benchmark problem that admits an analytical solution is used to assess the accuracy and computational efficiency of the numerical model. The results show that the flux limited second order temporal scheme substantially reduce numerical diffusion on relatively coarse meshes. The low temperature wood drying non-linear case study highlights that the first order temporal scheme combined with flux limiting achieves good accuracy on a relatively coarse mesh and improves the overall computational efficiency.

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