Forward Difference Method Research Articles

Efficient Jacobian evaluation method in multimode hierarchical circuit simulation

An efficient method for Jacobian evaluations in multimode hierarchical circuit simulation is presented. The method evaluates the Jacobian matrix based on the block Jacobian contributions which can be evaluated by various methods for different types of blocks to achieve both accuracy and efficiency. Comparison of the method with the forward-difference method shows considerable improvement in simulation efficiency and convergence.

Time-step sequences for parabolic differential equations

The Euler forward difference method is used for the explicit time integration of stiff systems of ordinary differential equations which originate from spatial discretization of parabolic partial differential equations. A sequence of non-equidistant time steps is derived. It primarily depends on one positive parameter, which determines an upper bound for the distance between the analytical solution and the approximation at every discrete time level in case the Jacobian matrix is constant. Use is made of modified Chebyshev polynomials. The asymptotic rate of convergence of the method is derived and its applicability in case the Jacobian matrix is non-constant is demonstrated.

Adsorptive Drying of Isopropyl Alcohol on 4A Molecular Sieves: Equilibrium and Kinetic Studies

Abstract Batch equilibrium and kinetic studies were carried out for adsorption of water from aqueous solution of isopropyl alcohol on 4A molecular sieves (cylindrical pellets of 1.8 mm diameter). Equilibrium data obtained were rectangular in shape. The adsorption capacity was about 0.25 g water per gram dry adsorbent. This corresponds to 0.29g water/cm3 of molecular sieve pellets. The effect of temperature on equilibrium behavior was not significant in the temperature range 20 to 70°C. Batch kinetic studies were carried out at room temperature. Differential equations for the uptake of water by biporous cylindrical particles were written and solved numerically by the finite forward difference method. The macropore diffusion coefficient was estimated by assuming the macroporosity of the pellet to be 0.35 and the value of tortuosity to be 3. The diffusion coefficient for micropores was obtained by matching the computed uptake curve with the experimental data. Assuming the average radius of microspheres to be 1 μm, the diffusion coefficient for micropores was found to be 0.4 × 10−12 cm2/s.

A numerical learning method for the vibration of a system with a single degree of freedom

A computer-based learning method for first-year undergraduates is described. The equation describing a vibrating system with a single degree of freedom is solved using the forward finite difference method with a variety of conditions.

A digital signal processing approach to real-time AC motor modeling

Discrete-time computational models of squirrel-cage AC motors are derived and analyzed for the purpose of advanced motor control. The starting point is a continuous-time model defined by a pair of simultaneous complex-coefficient differential equations. Discrete-time models are derived using both the bilinear transformation and the forward-difference approximation. The exact dependence between stability and the minimum sampling rate in the forward-difference approach is shown. The responses of the two discretized models are compared using real-world motor parameters. Using the forward difference method, a more compact implementation is obtained (although a higher sampling rate is required) as compared to the bilinear transformed model of equivalent performance and 2.7 times higher computational complexity. A 16/24-b DSP-ASIC coprocessor prototype implementing the AC motor model is introduced. >

Stability conditions when using theforward difference method

The forward difference method, also known as Euler's method, generates discretetime transfer functions that lead to compact implementations that are desirable for several industrial control applications. However, the potential of Euler's method is usually overlooked because some control literature gives misleading information on the stability of the z-domain poles obtained by Euler's method. In this paper, we derive analytical expressions for the minimum sampling rate that guarantees stability, and the optimum sampling rate that minimizes the modulus of a stable z-domain pole.

Numerical analysis of transient states in one‐dimensional non‐linear systems

AbstractAn algorithm for the analysis of non‐linear partial differential equations describing transient states of one‐dimensional physical problems with boundary conditions given on the opposite ends of the analysed domain is presented. The algorithm makes use of the Runge‐Kutta and bisection methods used in space ℝ1 and the forward finite difference method for the time domain. The algorithm is illustrated by one‐dimensional analysis of a temperature field in a transient state.

Thermal Bridges: A Non-Computerized Calculation Procedure

This study presents a non-computerized calculation procedure for steady state two-dimensional conductive heat losses through thermal bridges. The heat flow is divided into distinct flow paths based on physical considerations. A resistance is derived analytically for each flow path. These resistances are combined into a net work. The network simulates the construction. Calculating the total resistance of the network gives the thermal resistance of the construction. The algorithm is compared with a finite forward difference method. The relative error on the linear thermal transmittance of the thermal bridges calculated does not exceed 10%. The absolute error on the mean surface temperature is approximately 1°C and independent of the type of construction. The minimum inside surface temperature can be estimated in only a few cases; the absolute error is approximately 0.5°C. The algorithm is devel oped mainly for designers. Its simplicity and the possibility it offers for quick and ac curate control calculations makes it an ideal design tool. It can be programmed in Lotus for specific applications of a designer. It is ideal for prevention of condensa tion, more so than predicting heat loss (or gain).

A general finite-difference formulation with application to Navier-Stokes equations

A general method for deriving a finite-difference equation from a partial differential equation with the use of local model solutions is presented. The method, which is based on the evaluation of local functions, is illustrated by applying to a time-dependent vorticity equation. The main advantage of the present approach is that there is no apparent stability limitation on grid size and time step. Time dependent Navier-Stokes equations are solved with the present formulation for a Couette flow with constant suction and a square cavity flow problems. Steady-state, convergent solutions are obtained with grid sizes and time steps much larger than those determined from the conventional stability criteria for the forward time central difference method. The local model solutions used in the present paper were designed to produce steady-state solution; care must be taken to interpret the transient solution obtained from the present model with large time step.

Simulation of a steady-state integrated human thermal system

The mathematical model of an integrated human thermal system is formulated. The system consists of an external thermal regulation device on the human body. The purpose of the device (a network of cooling tubes held in contact with the surface of the skin) is to maintain the human body in a state of thermoneutrality. The device is controlled by varying the inlet coolant temperature and coolant mass flow rate. The differential equations of the model are approximated by a set of algebraic equations which result from the application of the explicit forward finite difference method to the differential equations. The integrated human thermal system is simulated for a variety of combinations of the inlet coolant temperature, coolant mass flow rate, and metabolic rates. Two specific cases are considered: (a) the external thermal regulation device is placed only on the head and (b) the devices are placed on the head and the torso. The results of the simulation indicate that when the human body is exposed to hot environment, thermoneutrality can be attained by localized cooling if the operating variables of the external regulation devices(s) are properly controlled.

摩擦項を含む渦度方程式の時間積分について

First, it is proved that the solution for frictional vorticity equation appears always computationally unstable, so far as the usual centered time difference method is used. As the substitute of the method above, another method to obtain a computationally stable and accurate solution is proposed. The principle is to apply the centered difference method only to the calculation of the advection term, and to apply the forward difference method in Taylor's series expansion form to that of the frictional term.