Lagrange Multiplier Method

Lagrange multipliers can be used to find the extreme of a multivariate function $f (x_{1}, x_{2}, ..., x_{n})$ subject to the constraint $g (x_{1}, x_{2}, ..., x_{n}) = 0$

Where, $f$ and $g$ are functions with continuous first partial derivatives on the open set containing the constraint curve, and $\nabla g \neq 0$ at any point on the curve (where $\nabla$ is the gradient).

To find the extreme, write:

図 1.

d f = \frac{\partial f}{\partial x_{1}} d x_{1} + \frac{\partial f}{\partial x_{2}} d x_{2} + ... + \frac{\partial f}{\partial x_{n}} d x_{n} = 0

But, because

g

is being held constant, it is also true that

図 2.

d g = \frac{\partial g}{\partial x_{1}} d x_{1} + \frac{\partial g}{\partial x_{2}} d x_{2} + ... + \frac{\partial g}{\partial x_{n}} d x_{n} = 0

So multiply 式 2 by the as yet undetermined parameter

λ

and add to 式 2,

図 3.

(\frac{\partial f}{\partial x_{1}} + λ \frac{\partial g}{\partial x_{1}}) d x_{1} + (\frac{\partial f}{\partial x_{2}} + λ \frac{\partial g}{\partial x_{2}}) d x_{2} + ... + (\frac{\partial f}{\partial x_{n}} + λ \frac{\partial g}{\partial x_{n}}) d x_{n} = 0

Note that the differentials are all independent, so any combination of them can be set equal to 0 and the remainder must still give zero. This requires that:

図 4.

(\frac{\partial f}{\partial x_{k}} + λ \frac{\partial g}{\partial x_{k}}) d x_{k} = 0

for all k = 1, ..., n, and the constant

λ

is called the Lagrange multiplier. For multiple constraints,

g_{1} = 0

g_{2} = 0

, ...,

図 5.

\nabla f = λ_{1} \nabla g_{1} + λ_{2} \nabla g_{2} + ...

The Lagrange multiplier method can be applied to contact-impact problems. In this case, the multivariate function is the expression of total energy subjected to the contact conditions:

図 6.

f (x_{1}, x_{2}, ..., x_{n}) \equiv Π (x, \dot{x}, \ddot{x})

図 7.

g (x_{1}, x_{2}, ..., x_{n}) \equiv Q (x, \dot{x}, \ddot{x}) = 0

Where,

x, \dot{x}, \ddot{x}

are the global vectors of DOF. The application of Lagrange multiplier method to the previous equations gives the weak form as:

図 8.

M \ddot{x} + f_{i n t} - f_{e x t} + L λ = 0

with

図 9.

L x = b

This leads to:

図 10.

[\begin{matrix} K & L^{T} \\ L & 0 \end{matrix}] {\begin{matrix} x \\ λ \end{matrix}} = {\begin{matrix} f \\ 0 \end{matrix}}

The Lagrange multipliers are physically interpreted as surface tractions. The equivalence of the modified virtual power principle with the momentum equation, the traction boundary conditions and the contact conditions (impenetrability and surface tractions) can be easily demonstrated. ^¹

It is emphasized that the above weak form is an inequality. In the discretized form, the Lagrange multiplier fields will be discretized and the restriction of the normal surface traction to be compressive will result from constraints on the trial set of Lagrange multipliers.

¹ Engelmann B.E. and Whirley R.G., A new elastoplastic shell element formulation for DYNA3D, Report ugrl-jc-104826, Lawrence Livermore National Laboratory, 1990.