Secant condition

Thron, C.D. 1991. The secant condition for instabihty in biochemical feedback control. I. The role of cooperativity and saturability. Bull. Math. BioL 53 383 1. 

Often we construct from past gradient evaluations. As the Hessian is the Jacobian of the gradient, we generate from so that it satisfies the secant condition... 

In modeling the SHG performanee of the bulk nonlinear optical crystals, we have assumed the incident fundamental pulses are characterized by a hyperbolic secant temporal profile, and therefore r = rg (FWHM)/l.76. For the case of KNhOs, the maximum optical-to-optical SHG efficiency achieved was 30%. The corresponding value of L/L, under these conditions, with a GVM parameter, 1.2 ps/mm, is L/L = 30. The values of beam... 

For the standard test pieces, force/deflection is recorded up to 25% at 10 mm/min, four cycles being made and from the fourth the secant moduli at 10% and 20% strain are obtained. The importance of mechanical conditioning has been discussed in Chapter 5 and this standard is one of the rare examples of where such a procedure is specified. Quite illogically, the test for products is normally done without mechanical conditioning and the force deflection curve is obtained to 30% so that the stiffness at 25% can be used as the result. Sensibly, provision would be made for the strain and the conditioning to be chosen to suit the particular application. 

Kic = coefficient of stress intensity under conditions of plane deformation max = coefficient of stress intensity under maximum fracture load Kq = coefficient of stress intensity under the load corresponding to 5% secant line Rff = notch radius... 

If the condition at x = 0 is not satisfied, a different value of a is chosen and the integration process is repeated until the above relation is finally satisfied within some prespecified error. This method is commonly called the shooting method. Use the secant method of Appendix A to show that the next guess of a should satisfy the following equation... 

If necessary, the prediction provided by the secant method can also be exploited on the condition that said prediction is checked it should be in the order of the current value of dt. 

Some of the worst conditions are met in excavations that have to be taken below the water table (Forth, 2004). In such cases, the water level must be lowered by some method of dewatering. The method adopted depends on the permeability of the ground and its variation within the stratal sequence, the depth of base level below the water table and the piezometric conditions in underlying horizons. Pumping from sumps within an excavation, bored wells or wellpoints are the dewatering methods most frequently used (Bell and Cashman, 1986). Impermeable barriers such as steel sheet piles, secant piles, diaphragm walls, frozen walls and grouted walls can be used to keep water out of excavations (Bell and Mitchell, 1986). Ideally, these structures should be keyed into an impermeable horizon beneath the excavation. 

The TG-secant method is a valuable tool for shortening the development and evaluation time and selecting the material composition and technological conditions, although this method is not as accurate as routine oven-aging testing. 

For the multivariable case, we need to consider additional conditions to obtain a secant step. Here, we define a matrix B that substitutes the Jacobian matrix and satisfies the secant relation ... 

Here Sy is the yield strength of the material at the mean value of the temperature of the cycle and is the value obtained from the applicable design fatigue curve for the specified number of cycles of the condition being considered. The use of Eq. (11.1) is recommended for local thermal stresses only. Such an approach with a modified expression for Poisson s ratio involving the secant modulus, Eg has been employed in low cycle fatigue evaluation by Gonyea and Moulin and Roche. In their assessments the Poisson s ratio expression is represented as... 

Tensile tests were performed, according to ISO 179 1993, in a Instron 1122 testing machine at standard conditions of temperature and humidity. From these tests the stress at maximum load, the strain at break and the secant modulus at 2 % were determined. 

In the PE method time-dependent property values for the modulus (include secant modulus) are selected and substituted into the standard equations. This approach is sufficiently accurate if the value chosen for the modulus takes into account the projected service life of the product and/or the limiting strain of the plastic. This approach is not a straightforward solution applicable to all plastics or even to one plastic in all its applications. This type of evaluation takes into consideration the value to use as a safety factor (SF). If no history exists a high value will be required. In time with service condition inputs, the SF can be reduced if justified (Chapter 7). 

In Figure 6.12 two different slopes have been guessed, resulting in two different solutions, neither of which match the boundary value Different root-finding algorithms may be used to find the missing initial condition, e.g. the secant method or the Newton method. When the missing initial condition has been found, the solution of the IVP y(b) will match... 

Cap In addition to the component of soil resistance acting on piles, the passive pressure soil resistance on the vertical pile cap face can be added to the stiffness and resistance obtained from the pile members. This is contingent on stable level ground conditions. The pile cap stiffness can be estimated as the ultimate soil capacity divided by an estimated displacement to mobilize this capacity. Centrifuge tests (Gadre 1997) showed that the deflection level to reach the ultimate pile cap capacity occurs at about 0.02 times the embedment depth. This equivalent linear secant stiffness can be added to the stiffness of the piles. 

Shown below is a MATLAB script (no function heading) file that prompts the user for two initial guesses for f(0). These guesses are used by the secant method to converge the right and boundary conditions, which is 7(1) = 0. 

Sapply the secant method to the right hand boundary condition... 

Exercise 10.6 Solve the problem described in Exercise 6.1 using MATLAB. Implement the secant method, as in Example 10.5, to converge the right-hand boundary condition. Use ode45 to solve the ODEs. 

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