2,866 research outputs found
Status of the differential transformation method
Further to a recent controversy on whether the differential transformation
method (DTM) for solving a differential equation is purely and solely the
traditional Taylor series method, it is emphasized that the DTM is currently
used, often only, as a technique for (analytically) calculating the power
series of the solution (in terms of the initial value parameters). Sometimes, a
piecewise analytic continuation process is implemented either in a numerical
routine (e.g., within a shooting method) or in a semi-analytical procedure
(e.g., to solve a boundary value problem). Emphasized also is the fact that, at
the time of its invention, the currently-used basic ingredients of the DTM
(that transform a differential equation into a difference equation of same
order that is iteratively solvable) were already known for a long time by the
"traditional"-Taylor-method users (notably in the elaboration of software
packages --numerical routines-- for automatically solving ordinary differential
equations). At now, the defenders of the DTM still ignore the, though much
better developed, studies of the "traditional"-Taylor-method users who, in
turn, seem to ignore similarly the existence of the DTM. The DTM has been given
an apparent strong formalization (set on the same footing as the Fourier,
Laplace or Mellin transformations). Though often used trivially, it is easily
attainable and easily adaptable to different kinds of differentiation
procedures. That has made it very attractive. Hence applications to various
problems of the Taylor method, and more generally of the power series method
(including noninteger powers) has been sketched. It seems that its potential
has not been exploited as it could be. After a discussion on the reasons of the
"misunderstandings" which have caused the controversy, the preceding topics are
concretely illustrated.Comment: To appear in Applied Mathematics and Computation, 29 pages,
references and further considerations adde
Minimum-lap-time optimisation and simulation
The paper begins with a survey of advances in state-of-the-art minimum-time simulation for road vehicles. The techniques covered include both quasi-steady-state and transient vehicle models, which are combined with trajectories that are either pre-assigned or free to be optimised. The fundamentals of nonlinear optimal control are summarised. These fundamentals are the basis of most of the vehicular optimal control methodologies and solution procedures reported in the literature. The key features of three-dimensional road modelling, vehicle positioning and vehicle modelling are also summarised with a focus on recent developments. Both cars and motorcycles are considered
The turnpike property in finite-dimensional nonlinear optimal control
Turnpike properties have been established long time ago in finite-dimensional
optimal control problems arising in econometry. They refer to the fact that,
under quite general assumptions, the optimal solutions of a given optimal
control problem settled in large time consist approximately of three pieces,
the first and the last of which being transient short-time arcs, and the middle
piece being a long-time arc staying exponentially close to the optimal
steady-state solution of an associated static optimal control problem. We
provide in this paper a general version of a turnpike theorem, valuable for
nonlinear dynamics without any specific assumption, and for very general
terminal conditions. Not only the optimal trajectory is shown to remain
exponentially close to a steady-state, but also the corresponding adjoint
vector of the Pontryagin maximum principle. The exponential closedness is
quantified with the use of appropriate normal forms of Riccati equations. We
show then how the property on the adjoint vector can be adequately used in
order to initialize successfully a numerical direct method, or a shooting
method. In particular, we provide an appropriate variant of the usual shooting
method in which we initialize the adjoint vector, not at the initial time, but
at the middle of the trajectory
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State-of-the-art on research and applications of machine learning in the building life cycle
Fueled by big data, powerful and affordable computing resources, and advanced algorithms, machine learning has been explored and applied to buildings research for the past decades and has demonstrated its potential to enhance building performance. This study systematically surveyed how machine learning has been applied at different stages of building life cycle. By conducting a literature search on the Web of Knowledge platform, we found 9579 papers in this field and selected 153 papers for an in-depth review. The number of published papers is increasing year by year, with a focus on building design, operation, and control. However, no study was found using machine learning in building commissioning. There are successful pilot studies on fault detection and diagnosis of HVAC equipment and systems, load prediction, energy baseline estimate, load shape clustering, occupancy prediction, and learning occupant behaviors and energy use patterns. None of the existing studies were adopted broadly by the building industry, due to common challenges including (1) lack of large scale labeled data to train and validate the model, (2) lack of model transferability, which limits a model trained with one data-rich building to be used in another building with limited data, (3) lack of strong justification of costs and benefits of deploying machine learning, and (4) the performance might not be reliable and robust for the stated goals, as the method might work for some buildings but could not be generalized to others. Findings from the study can inform future machine learning research to improve occupant comfort, energy efficiency, demand flexibility, and resilience of buildings, as well as to inspire young researchers in the field to explore multidisciplinary approaches that integrate building science, computing science, data science, and social science
Physics-informed Neural Networks approach to solve the Blasius function
Deep learning techniques with neural networks have been used effectively in
computational fluid dynamics (CFD) to obtain solutions to nonlinear
differential equations. This paper presents a physics-informed neural network
(PINN) approach to solve the Blasius function. This method eliminates the
process of changing the non-linear differential equation to an initial value
problem. Also, it tackles the convergence issue arising in the conventional
series solution. It is seen that this method produces results that are at par
with the numerical and conventional methods. The solution is extended to the
negative axis to show that PINNs capture the singularity of the function at
$\eta=-5.69
A new scalable algorithm for computational optimal control under uncertainty
We address the design and synthesis of optimal control strategies for
high-dimensional stochastic dynamical systems. Such systems may be
deterministic nonlinear systems evolving from random initial states, or systems
driven by random parameters or processes. The objective is to provide a
validated new computational capability for optimal control which will be
achieved more efficiently than current state-of-the-art methods. The new
framework utilizes direct single or multi-shooting discretization, and is based
on efficient vectorized gradient computation with adaptable memory management.
The algorithm is demonstrated to be scalable to high-dimensional nonlinear
control systems with random initial condition and unknown parameters.Comment: 23 pages, 17 figure
Effects of radiation and chemical reaction on convective nanofluid flow through a non-linear permeable stretching sheet with partial slip
The steady, convective two-dimensional nanofluid flow has been investigated under the influence of radiation absorption and chemical reaction through a porous medium. The flow has been caused by a non-linear stretching sheet with the slip effects of the velocity, the temperature and the nanoparticle concentration. The fluid is electrically conducted in the presence of an applied magnetic field. Appropriate transformations reduce the non-linear partial differential system to an ordinary differential system. The convergent solutions of the governing non-linear problems have been computed using fifth-order-Runge-Kutta-Fehlberg integration scheme. The results of the velocity, the temperature, and the concentration fields have been calculated in series forms. The effects of the different parameters on the velocity, the temperature, and the concentration profiles are shown and analyzed. The skin friction coefficient, the Nusselt and the Sherwood numbers have also been computed and investigated for different embedded parameters in the problem statement
Numerical study of magneto-convective heat and mass transfer from inclined surface with Soret diffusion and heat generation effects : a model for ocean magnetohydrodynamics energy generator fluid dynamics
A mathematical model is developed for steady state magnetohydrodynamic (MHD) heat and
mass transfer flow along an inclined surface in an ocean MHD energy generator device with
heat generation and thermo-diffusive (Soret) effects. The governing equations are
transformed into nonlinear ordinary differential equations with appropriate similarity
variables. The emerging two-point boundary value problem is shown to depend on six
dimensionless thermophysical parameters - magnetic parameter, Grashof number, Prandtl
number, modified Prandtl number, heat source parameter and Soret number in addition to
plate inclination. Numerical solutions are obtained for the nonlinear coupled ordinary
differential equations for momentum, energy and salinity (species) conservation, numerically,
using the Nachtsheim-Swigert shooting iteration technique in conjunction with the Runge-
Kutta sixth order iteration scheme. Validation is achieved with Nakamura’s implicit finite
difference method. Further verification is obtained via the semi-numerical Homotopy
analysis method (HAM). With an increase in magnetic parameter, skin friction is depressed
whereas it generally increases with heat source parameter. Salinity magnitudes are
significantly reduced with increasing heat source parameter. Temperature gradient is
decreased with Prandtl number and salinity gradient (mass transfer rate) is also reduced with
modified Prandtl number. Furthermore, the flow is decelerated with increasing plate
inclinations and temperature also depressed with increasing thermal Grashof number
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