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# Question on the definitions of secant stiffness and tangent stiffness in iterative methods for non-linear problems

Recently, I find that definitions of secant stiffness and tangent stiffness in many books seem pretty confused.Therefore, here I am giving the definitions I think correct, then give my questions on them.

Consider the equation

H(x)x+f=0

(1) Definition of tangent stiffness. As for the above equation, what is the tangent stiffness for some particular value x0?

Let's give the definition according basic knowledge of advanced mathematics. Let's consider the curve H(x)x~x. For this curve, y=H(x)x, so the tangent at x0 is

HT=dy/dx=d[H(x)x]/dx=H(x) + x dH/dx

It can be imagined that generally HT is not symmetric.

(2) Definition of secant stiffness. We imagine from the term "secant", that two points are needed to define a secant stiffness. If the 1st point is (0,0) in the diagram of H(x)x~x, and the second point is (x, H(x)x), then the secant between these two points is

HS=[H(x)x-0]/(x-0)=H(x)

Therefore, H(x) is NOT the tangent stiffness, BUT the secant stiffness between the above two points.

However, in section 2.2.3 Owen's book "plastic finite element--theory and application", H(x) is taken as so-called tangent stiffness.

Above is my understanding about these two definitions. Can anyone tell me whether they are right? Please tell us sth on this topic :)

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