Corrected some minor typos.

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Craig Warren
2016-06-09 14:26:38 +01:00
父节点 98493fdbc7
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@@ -20,7 +20,7 @@ All electromagnetic phenomena, on a macroscopic scale, are described by the well
where :math:`t` is time (seconds) and :math:`q_v` is the volume electric charge density (coulombs/cubic metre). In Maxwell's equations, the field vectors are assumed to be single-valued, bounded, continuous functions of position and time. In order to simulate the GPR response from a particular target or set of targets the above equations have to be solved subject to the geometry of the problem and the initial conditions.
The nature of the GPR forward problem classifies it as an *initial value -- open boundary* problem. This means that in order to obtain a solution you have to define an initial condition (i.e. excitation of the GPR transmitting antenna) and allow for the resulting fields to propagate through space reaching a zero value at infinity since, there is no specific boundary which limits the problem's geometry and where the electromagnetic fields can take a predetermined value. Although the first part is easy to accommodate (i.e. specification of the source), the second part can notbe easily tackled using a finite computational space.
The nature of the GPR forward problem classifies it as an *initial value -- open boundary* problem. This means that in order to obtain a solution you have to define an initial condition (i.e. excitation of the GPR transmitting antenna) and allow for the resulting fields to propagate through space reaching a zero value at infinity since, there is no specific boundary which limits the problem's geometry and where the electromagnetic fields can take a predetermined value. Although the first part is easy to accommodate (i.e. specification of the source), the second part cannot be easily tackled using a finite computational space.
The FDTD approach to the numerical solution of Maxwell's equations is to discretize both the space and time continua. Thus the discretization spatial :math:`\Delta x`, :math:`\Delta y` and :math:`\Delta z` and
temporal :math:`\Delta t` steps play a very significant role -- since the smaller they are the closer the FDTD model is to a real representation of the problem. However, the values of the discretization steps always have to be finite, since computers have a limited amount of storage and finite processing speed. Hence, the FDTD model represents a discretized version of the real problem and is of limited size. The building block of this discretized FDTD grid is the Yee cell [YEE1966]_ named after Kane Yee who pioneered the FDTD method. This is illustrated for the 3D case in :numref:`yeecell`.
@@ -36,7 +36,7 @@ By assigning appropriate constitutive parameters to the locations of the electro
The numerical solution is obtained directly in the time domain by using a discretized version of Maxwell's curl equations which are applied in each FDTD cell. Since these equations are discretized in both space and time the solution is obtained in an iterative fashion. In each iteration the electromagnetic fields advance (propagate) in the FDTD grid and each iteration corresponds to an elapsed simulated time of one :math:`\Delta t`. Hence by specifying the number of iterations you can instruct the FDTD solver to simulate the fields for a given time window.
The price you has to pay of obtaining a solution directly in the time domain using the FDTD method is that the values of :math:`\Delta x`, :math:`\Delta y`, :math:`\Delta z` and :math:`\Delta t` can not be assigned independently. FDTD is a conditionally stable numerical process. The stability condition is known as the CFL condition after the initials of Courant, Freidrichs and Lewy and is given by,
The price you have to pay for obtaining a solution directly in the time domain using the FDTD method is that the values of :math:`\Delta x`, :math:`\Delta y`, :math:`\Delta z` and :math:`\Delta t` can not be assigned independently. FDTD is a conditionally stable numerical process. The stability condition is known as the CFL condition after the initials of Courant, Freidrichs and Lewy and is given by,
.. math:: \Delta t \leq \frac{1}{c\sqrt{\frac{1}{(\Delta x)^2}+\frac{1}{(\Delta y)^2}+\frac{1}{(\Delta z)^2}}},