Inverse analysis for parameter estimation of sandy soil with axially loaded pile using nonlinear programming

Abstract

A computerized mathematical procedure based on nonlinear programming is presented for the purpose of a sandy soil parameter prediction problem by inverse analysis. The task of inverse analysis is to evaluate the values of input parameters for a specified output or response of the system. This inverse problem to determine the values of system input parameters is mathematically formulated as a constrained optimization problem in this study. The input parameters are considered as design variables. The constraint set is developed through the implementation of lower and upper bound on design variables. The objective function is determined by evaluating the error or mismatch between the specified output (reference value) and the model predicted value for a given design vector. The resulting nonlinear programming problem is solved using the Interior Penalty Function method coupled with the Davidon-Fletcher-Powell method and quadratic interpolation scheme. To demonstrate the proposed methodology, two illustrative examples related to axially loaded piles installed in sandy soil medium are considered. The numerical results indicate that the back analyzed values are in good agreement with their respective reference values.

Appendix A

Mathematical method to solve a nonlinear programming (NLP) problem

A standard NLP problem may be defined as:

Determine a vector of design variables $\vec{X}$ that minimizes the objective function $f(\vec{X})$ satisfying the set of constraint

$$ g_{j} (\vec{X}) \le 0,j = 1,2,\ldots,m $$

(A1)

To solve the NLP problem, IPF based method coupled with the DFP method and QIM is used in this study. These three methods are used for the following purposes.

IPF: constrained optimization problem is converted into an equivalent unconstrained optimization problem.

DFP: minimizes the unconstrained problem.

QIM: calculation of optimum step length along the direction of search from a given point.

1.1 IPF method

Generate a new function $\phi (\vec{X},r_{k} )$ as

$$ \phi (\vec{X},r_{k} ) = f(\vec{X}) - r_{k} \sum\limits_{j = 1}^{m} {\frac{1}{{g_{j} (\vec{X})}}} $$

(A2)

where, r_k is the positive constant (also called as penalty parameter). In the general initial value of r_k is taken as such two terms in the right-hand side of equation (A2) are the same at the initial feasible design point $\vec{X}_{0}$.

Starting value of penalty parameter

$$ r = r_{1} = - \frac{{f(\vec{X}_{0} )}}{{\sum\limits_{j} {\frac{1}{{g_{j} (\vec{X}_{0} )}}} }} $$

(A3)

1.2 DFP method

For r = r₁, $\phi$ function is minimized beginning from the design point $\vec{X}_{0}$ according to the DFP method. This involves the following steps:

• (1) At a point $\vec{X}_{0}$, generate a direction of movement $\vec{S}_{0}$ as

  $$ S_{0} = - \left[ {H_{0} } \right]\nabla \phi_{0} $$

  (A4)

[H₀] is an initial symmetric positive definite matrix. Initially, it may be assumed as an identity matrix. $\nabla \phi_{0}$ is gradient vector of $\phi$ function calculated at $\vec{X}_{0}$. $S_{0}$ is known as the direction of search vector evaluated at $\vec{X}_{0}$.

• (2) Determine a new vector $\vec{X}_{i + 1}$ as

  $$ \vec{X}_{i + 1} = \vec{X}_{i} + \lambda_{i}^{*} \vec{S}_{i} $$

  (A5)

here, $\lambda_{i}^{*}$ minimizes $\phi (X_{i} + \lambda_{i}^{*} S_{i} )$, it is known as one-dimensional minimization. $\lambda_{i}^{*}$ is known as optimum step length along the direction of search $\vec{S}_{i}$ and the quadratic interpolation method [28] is used to evaluate the optimum step length.

• (3) Evaluate $\nabla \phi_{i + 1}$, that indicates the gradient of $\phi$ at $\vec{X}_{i + 1}$.

At $\vec{X}_{i + 1}$, the new direction of movement is obtained as

$$ \vec{S}_{i + 1} = - \left[ {H_{i + 1} } \right]\nabla \phi_{i + 1} $$

(A6)

where

$$ [H_{i + 1} ] = [H_{i} ] + [M_{i} ] + [N_{i} ] $$

(A7)

with

$$ \left[ {M_{i} } \right] = \lambda_{i}^{*} \frac{{S_{i} S_{i}^{T} }}{{S_{i}^{T} Y_{i} }} $$

(A8)

$$ \vec{Y}_{i} = \nabla \phi_{i + 1} - \nabla \phi_{i} $$

(A9)

and

$$ \left[ {N_{i} } \right] = - \frac{{P_{i} P_{i}^{T} }}{{Y_{i}^{T} P_{i} }} $$

(A10)

$$ P_{i} = [H_{i} ]\vec{Y}_{i} $$

(A11)

• (4) Once the new direction of movement $\vec{S}_{i + 1}$ is evaluated, continue steps (2) through (4) until the function $\phi$ is minimized at $X = X_{i + 1}^{*}$.

• (5) Set a new value of r that is equal to 1/10 of its present value and generate a new function $\phi$ with the new value of r. Taking the end point of step (4) as a new starting point and repeating the above steps $\phi (\vec{X},r_{k} )$ is minimized. This is continued until the specified convergence condition is satisfied.

1.3 QIM

The optimum step length ($\lambda_{i}^{*}$) along the search direction $\vec{S}_{i}$ from the present point $\vec{X}_{i}$ is evaluated by the quadratic interpolation method. In this method, three values of step length ($\lambda$) are chosen in a specific manner. A quadratic equation h($\lambda$) is fit between these three points as an approximation of the actual curve $\phi (\lambda )$. The value of optimum step length along the search direction is obtained by differentiating h($\lambda$) with respect to $\lambda$ and its closed-form expression can be obtained in terms of the $\phi$ function value at three chosen points. Major steps are given below.

The function $\phi (\lambda )$ is approximated by a quadratic function h($\lambda$) which has an easily determinable minimum point. h($\lambda$) is expressed as:

$$ h(\lambda ) = a + b\lambda + c\lambda^{2} $$

(A12)

the minimum of which occurs when

$$ \frac{dh}{{d\lambda }} = b + 2c\lambda^{*} = 0.0 $$

(A13)

$$ \lambda^{*} = \frac{ - b}{{2c}} $$

(A14)

The constants a, b and c of the approximating quadratic can be determined by sampling the function at three different $\lambda$ values, $\lambda_{1}$, $\lambda_{2}$ and $\lambda_{3}$ and solving the equations:

$$ \phi_{1} = a + b\lambda_{1} + c\lambda_{1}^{2} $$

(A15)

$$ \phi_{2} = a + b\lambda_{2} + c\lambda_{2}^{2} $$

(A16)

$$ \phi_{3} = a + b\lambda_{3} + c\lambda_{3}^{2} $$

(A17)

where, $\phi_{1}$ denotes $\phi_{1} (\lambda_{1} )$ and so on.

The constants a, b and c can be determined as follows.

$$ c = \frac{{\left[ {\left( {\phi_{3} - \phi_{1} } \right)\left( {\lambda_{2} - \lambda_{1} } \right) - \left( {\phi_{2} - \phi_{1} } \right)\left( {\lambda_{3} - \lambda_{1} } \right)} \right]}}{{\left[ {\left( {\lambda_{3} - \lambda_{1} } \right)\left( {\lambda_{3} - \lambda_{2} } \right)\left( {\lambda_{2} - \lambda_{1} } \right)} \right]}} $$

(A18)

$$ b = \left[ {\left( {\phi_{2} - \phi_{1} } \right)/\left( {\lambda_{2} - \lambda_{1} } \right)} \right] - c\left( {\lambda_{2} + \lambda_{1} } \right) $$

(A19)

$$ a = \phi_{1} - b\lambda_{1} - c\lambda_{1}^{2} $$

(A20)

Further details of this method have been elaborated in standard textbooks on optimization techniques [28].