Lookup table

Cellsim s universe consists of a 512 x 512 array. The program itself is completely menu-driven and includes these features (1) 256-state computed-function rules, in addition to lookup-tables, on both the Sun and CM (2) creation of look-up tables using Langton s lambda parameter (see section 3.2) (3) the ability to save images in Sun raster format and (4) the ability to generate lookup tables with standalone C code. Cellsim can be obtained via anonymous ftp on the Internet to think.com. 

An area of development is the inclusion of code necessary for free energy perturbation calculations for internal, van der Waals and electrostatic energy terms ( ). Another area of development is the option to use lookup tables for both nonbond energies and for constraint energy terms such as a solvent boundary energy term (16). Other features and options are routinely added as needed to aid in the execution of current projects underway within the NIH. 

In a fluid model the correct calculation of the source terms of electron impact collisions (e.g. ionization) is important. These source terms depend on the EEDF. In the 2D model described here, the source terms as well as the electron transport coefficients are related to the average electron energy and the composition of the gas by first calculating the EEDF for a number of values of the electric field (by solving the Boltzmann equation in the two-term approximation) and constructing a lookup table. 

Dispersion model calculations are normally used to estimate downwind concentrations these concentrations are the basis for determining the consequences resulting from toxicity, fires, and/or explosions. For those not interested in using dispersion models, the standard includes lookup tables for all the listed substances to help a facility determine the impact distances for specific release scenarios. 

Consequence-Based Ranking Systems Release consequence modeling can be used to rank potential chemical hazards. For example, the USEPAs RMP regulations require consequence modeling for a predefined worst-case scenario—release of the entire contents of the largest container of a material in 10 min. EPA provides lookup tables and software (RMPComp) to assist in estimating the hazard distances for materials covered by the RMP regulations. 

Figure 5.10. Sketch of one-dimensional mixture-fraction chemical lookup table. For any value of the mixture fraction, the reacting scalars can be found from the pre-computed table in a post-processing stage of the flow calculation.

Note that the numerical simulation of the turbulent reacting flow is now greatly simplified. Indeed, the only partial-differential equation (PDE) that must be solved is (5.100) for the mixture-fraction vector, which involves no chemical source term Moreover, (5.151) is an initial-value problem that depends only on the inlet and initial conditions and is parameterized by the mixture-fraction vector it can thus be solved independently of (5.100), e.g., in a pre(post)-processing stage of the flow calculation. For a given value of , the reacting scalars can then be stored in a chemical lookup table, as illustrated in Fig. 5.10. 

Figure 5.11. Chemical lookup table parameterized in terms of mixture-fraction mean X = (f)... 

In the equilibrium-chemistry limit, the turbulent-reacting-flow problem thus reduces to solving the Reynolds-averaged transport equations for the mixture-fraction mean and variance. Furthermore, if the mixture-fraction field is found from LES, the same chemical lookup tables can be employed to find the SGS reacting-scalar means and covariances simply by setting x equal to the resolved-scale mixture fraction and x2 equal to the SGS mixture-fraction variance.88... 

Because x appears as a parameter in the flamelet model, in numerical implementations a flamelet library (Pitsch and Peters 1998 Peters 2000) is constructed that stores T( , x ) forO < < 1 in a lookup table parameterized by ( ), (4, 2),and x - Based on the definition of a flamelet, at any point in the flow the reaction zone is assumed to be isolated so that no interaction occurs between individual flamelets. In order for this to be true, the probabilities of finding f = 0 and f = 1 must both be non-zero. 

In theory, an arbitrary number of scalars could be used in transported PDF calculations. In practice, applications are limited by computer memory. In most applications, a reaction lookup table is used to store pre-computed changes due to chemical reactions, and models are limited to five to six chemical species with arbitrary chemical kinetics. Current research efforts are focused on smart tabulation schemes capable of handling larger numbers of chemical species. 

Fractional time stepping is widely used in reacting-flow simulations (Boris and Oran 2000) in order to isolate terms in the transport equations so that they can be treated with the most efficient numerical methods. For non-premixed reactions, the fractional-time-stepping approach will yield acceptable accuracy if A t r . Note that since the exact solution to the mixing step is known (see (6.248)), the stiff ODE solver is only needed for (6.249), which, because it can be solved independently for each notional particle, is uncoupled. This fact can be exploited to treat the chemical source term efficiently using chemical lookup tables. 

In a transported PDF simulation, the chemical source term, (6.249), is integrated over and over again with each new set of initial conditions. For fixed inlet flow conditions, it is often the case that, for most of the time, the initial conditions that occur in a particular simulation occupy only a small sub-volume of composition space. This is especially true with fast chemical kinetics, where many of the reactions attain a quasi-steady state within the small time step At. Since solving the stiff ODE system is computationally expensive, this observation suggests that it would be more efficient first to solve the chemical source term for a set of representative initial conditions in composition space,156 and then to store the results in a pre-computed chemical lookup table. This operation can be described mathematically by a non-linear reaction map ... 

In other words, a pre-computed lookup table must cover the entire allowable region, while only a small sub-set (the accessed region) is used in a particular simulation. 

Due to table-storage limitations, the applicability of pre-computed lookup tables will be limited by the dimensions of the allowable region. Standard pre-computed lookup tables (i.e., ones that do not attempt to find a low-dimensional representation of the chemical kinetics) will be limited by computer memory to three to five chemical species. For example, five scalars on a reasonably refined grid yields ... 

Since the quantity of real numbers that must be stored in the table will increase as lO2, it should be obvious to the reader that detailed kinetics schemes (which often involve tens or even hundreds of species) cannot be treated using pre-computed lookup tables. Considerable effort has thus been directed towards smart tabulation algorithms. 

In situ adaptive tabulation (IS AT) was proposed by Pope (1997), and it overcomes many of the difficulties associated with pre-computed lookup tables. First, the in situ nature of the method reduces the tabulation to only those points that occur during a particular simulation (i.e., the accessed region). Secondly, an adaptive algorithm is employed to control interpolation errors while minimizing the number of points that must be tabulated. 

Thirdly, the binary-tree tabulation algorithm used in ISAT is very different from the grid-based method described above for pre-computed lookup tables. We will look at each of these aspects in detail below. However, we will begin by briefly reviewing a few points from non-linear systems theory that will be needed to understand ISAT. 

Using particle partitioning, particle codes exhibit excellent scalability on distributed computing platforms (i.e., cluster computers). However, with complex chemistry, care must be taken when implementing chemical lookup tables to avoid scale-up bottlenecks. 

However, in practice, the integral must be evaluated using a stiff ODE solver or chemical lookup tables (see Section 6.9). Because transported PDF simulations are typically used for reacting flows with complex chemistry, the chemical-reaction step will often dominate the overall computational cost. It is thus important to consider carefully the computational efficiency of the chemical-reaction step when implementing a transported PDF simulation. 

Because the spectral irradiance 1(1) enters in this definition, the correction factor is a function of total ozone (Q) and solar zenith angle (6), which are the main parameters that determine 1(1). It is obvious from (2), that /ri expresses actually the relative change in the weighted integral when the instrument s sensitivity is used as weighting function instead of the CIE, Because of the involvement of 1(1) in relative sense, 1(1) can be determined from model calculations, as far as total ozone and SZA are known. In practice, lookup tables can be constructed for each instrument, defining the correction factor fd, as functions of total ozone and SZA. These tables can be used thereafter for the conversion of the broadband detectors radiometric output to CIE weighted eiythemal irradiance. 

Low-noise oscillator synthesis is not trivial however most methods use lookup tables with fractional interpolation. Oscillators can be implemented by (a) table lookup or (b) HR filters with poles located exactly on the unit circle. 

Moore [Moore, 1977b] studied the effect of oscillator implementation using lookup tables and found that linear interpolation produces the least distortion and that truncation produces the worst. This result was confirmed by Hartmann [Hartmann, 1987], Another possibility is to use a recursive (HR) filter with poles located on the unit circle. This coupled form [Tierney et al., 1971] offers a alternate method that avoids using memoiy space. Frequency resolution requirements were calculated by Snell in a superpipeline oscillator design for dynamic Fourier synthesis [Snell, 1977]. 

Linear Synthesis. The most popular method of synthesis is so-called Additive Synthesis , where the output is a sum of oscillators. While it is commonly assumed that the oscillators produce sinusoids (Fourier synthesis), in fact, they can be any waveform. Furthermore, with static additive synthesis, a pre-mixed combination of harmonics was stored in the lookup table. Unfortunately, this doesn t permit inharmonic partials. Dynamic Fourier synthesis allows the amplitudes and frequencies of the partials to be varied relative to each other. Computationally, it is important to recognize the that updating oscillator coefficients for large numbers of oscillators can be expensive. 

For operation in fixed-point DSP chips, the independent variable h = pm/2 — pjj" is generally confined to the interval [-1, 1). Note that having the table go all the way to zero at the maximum negative pressure hA+ = -1 is not physically reasonable (0.8 would be more reasonable), but it has the practical benefit that when the lookup-table input signal is about to clip, the re flection coefficient goes to zero, thereby opening the feedback loop. 

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