China 
Africa 
Explore chapters and articles related to this topic
Review of Basic Laws and Equations
Published in Pradip Majumdar, Computational Methods for Heat and Mass Transfer, 2005
Truncation errors are those that result from using an approximation in place of exact mathematical operations and quantities. The mathematical statement of fluid flow and heat transfer problems usually involves derivatives or integrals of different orders. Approximations to such quantities are obtained by retaining a certain number of terms in a power series instead of using an infinite number of terms to obtain exact results. Let us consider the problem in which a quantity is approximated by a truncated power series. If a function ϕ(x) and its n+1 derivatives are continuous on an interval xi+1,xi, then the value of the function at xi+1 is given in terms of the function value and its derivatives at another close point xi by the Taylor series ϕxi+1=ϕxi+ϕ'xih+ϕ''xi2!h2+ϕ'''xi3!h3+⋯+ϕ(n)xin!hn+Rn
Optimal local truncation error method for solution of 2-D elastodynamics problems with irregular interfaces and unfitted Cartesian meshes as well as for post-processing
Published in Mechanics of Advanced Materials and Structures, 2023
As can be seen, the presented procedure provides the third order of the local truncation error for the 9-point uniform stencils with the general geometry of the interface. The 9-point uniform stencils of OLTEM for homogeneous materials (without interface) provide the fourth order of the local truncation error. This leads to the second order of accuracy of global solutions; see the numerical examples below. Moreover, due to the minimization of the leading high-order terms of the local truncation error with Eqs. (28) and (29), at the same numbers of degrees of freedom and at the engineering accuracy, OLTEM with irregular interfaces yields more accurate results than those obtained by high-order finite elements (up to the third order) with much wider stencils; see the numerical examples below.
Improved models of solar radiation and convective heat transfer for pavement temperature prediction
Published in International Journal of Pavement Engineering, 2022
The global energy balance models and heat transfer models were implemented in Matlab™, and a one-dimensional finite element model was developed for each of the pavements in the SMP data. As discussed in the previous section, the EICM uses a one-dimensional finite difference method to solve the heat transfer model. Gray and Pinder (1976) detail the mathematical relationship between finite element and finite difference methods, noting that they are both used (along with finite volume methods) to solve the same class of problems. However, the truncation errors in the finite difference methods are very sensitive to the selected geometry, time steps and other considerations, which means that increasing the number of nodes analysed in the pavement segment must be compensated for by reducing the time steps to control the truncation errors. Finite element methods on the other hand are much more stable, but also considerably more computationally expensive. This research used finite elements with the same boundary conditions and partial differential equation as the EICM (Equations (14) and (15)), which allowed more flexibility with defining the nodal geometry to obtain more detailed results while maintaining numerical stability.
Extended formulations of lower-truncated transversal polymatroids
Published in Optimization Methods and Software, 2021
Hiroshi Imai, Keiko Imai, Hidefumi Hiraishi
Edmonds introduced a polymatroid as a polytope in his seminal paper [1] by using the lower truncation from its beginning, and we use his terminology below in this section to pay respect to the paper. A set function is a -function if it satisfies the following: (1) for , (2) for (monotonicity), (3) for (submodularity). Then, a polytope is a polymatroid, where .