APPLICATION OF THE FINITE ELEMENT METHOD TO SOLVING THE DUFFING EQUATION OF GROUND MOTION

In this paper, we applied the Galerkin Finite Element Method to solve a damped, externally forced, second order ordinary differential equation with cubic nonlinearity known as the Duffing Equation. The Galerkin method uses the functional minimization technique which sets the equation in systems of algebraic equations to be solved. Various simulation on the effect of change on some parametric values of the Duffing equation are shown.


The
Finite Element Method (FEM) is a numerical/computational analysis tool used in obtaining approximate solutions to boundary value problems which are governed by a differential equation and a set of boundary conditions. The main idea behind the finite element method is the representation of the domain with smaller subdomains called the finite elements. The distribution of the primary unknown quantity inside an element is then interpolated based on the values at the nodes, so far as nodal elements are used. The interpolation or shape functions must also be a complete set of polynomials. The accuracy of the solution depends, among other factors, on the order of these polynomials, whose order may be linear, quadratic, or higher. The numerical solution corresponds to the values of the primary unknown quantity at the nodes or the edges of the discretized domain. The solution is obtained after solving a system of linear equations. To form such a linear system of equations, the governing differential equation and associated boundary conditions must first be converted to an integro-differential form by using a weighted residual method as the Galerkin approach. This integro-differential formulation is applied to a single element and with the use of proper weight and interpolation functions called the shape functions, the respective element equations are obtained. The assembly of all elements results in a global matrix system that represents the entire domain of the BVP. The FEM was first used by  and it was invented when aircraft engineers wanted to solve problems relating to structures, wheels and to calculate stress in structures. Since then, the FEM has been used by many researchers in areas such as Electromagnetics [3 -5]

THE DUFFING EQUATION
The Duffing Oscillator or equation is a damped, externally forced, second order nonlinear oscillator with constant coefficients which has gained wide recommendations as the simplest equation which is used to study and describe the chaotic behaviour of a system. It is a nonlinear differential equation which describes an oscillator with a cubic nonlinearity [11]. It was developed by a German Engineer named Georg Wilhem Christian Caspar Duffing in 1918 who aimed to tackle problems of nonlinear oscillators in a systematic way by starting with the linear oscillator and also examining the effects of quadratic and cubic stiffness nonlinearities. He then emphasized the differences between the linear and nonlinear oscillators for both the free and forced vibrations while considering their damping effects. As a dynamical system that exhibits chaotic behaviour, It has a periodic long term behaviour as t → ∞ making the system to have irregular pattern which does not oscillate nor repeat in a periodic manner. It is always vigilant in the change of its initial conditions as any change will alter the trajectory thereby giving a significant difference in its long term behaviour.
The nonlinearity of the system makes it to exhibit irregular behaviour hence making it to be deterministic.
The general Duffing equation is given as where the unknown function ( ) x x t = is the displacement at time t . The damping factor δ controls the size of the damping. The α controls the size of the stiffness and the β controls the amount of nonlinearity in the restoring force. If 0 β = , the Duffing equation describes a damped and driven simple harmonic oscillator. The quantity γ controls the amplitude of the periodic driving force. If 0 γ = , we have a system without driving force. The quantity ω controls the frequency of the periodic driving force. See [12 -14]. When the Duffing equation has a negative linear stiffness, it is said to be a double-well potential. See [15 -17]. The Duffing double-well potential equation is given as

THE FINITE ELEMENT METHOD
The Finite Element Method is an element wise application of the weighted residual or Galerkin method which involves the following steps: a. Discretization of the domain b.
Formulation or derivation of element equations c.
Assembly of the element equations d.
Imposition of the boundary conditions e.

APPLICATION:
Consider the given Duffing oscillator, xሷ + δxሶ + αx + βx ଷ = γcoswt (3) Subject to xሺ0ሻ = 0, xሶ ሺ1ሻ = l We seek for an approximate solution of the form The error or residue E is given as Step 1: Discretization of the Domain Let the function be divided into four elements in the domain ሾ0,1ሿ. Let w = wሺtሻ be the test function and let s ଵ ands ଶ be successive node points. For each element e ୧ , i = 1, 2, 3, 4, we have two nodes and from ( ‫‬ ሺxሷ + δxሶ + αx + βx ଷ ሻwdt Step 2: Derivation of element equations Let where i ψ and j ψ are shape functions; i γ and i α are coefficients for the particular node which is 1 and 0 at other But s ଵ = s ୧ , s ଶ = s ୧ାଵ , wheni = 1 andh = s ୧ାଵ − s ୧ Substituting (7,8) into (6), we obtain Equation (9) can be written as where k ୧୨ ୣ is the stiffness matrix and it is given as Then, from (11) Hence, for a typical element, its stiffness matrix is 11 12 Step 3: Assembling of element equations For every element, The subscripts 1 and 2 are nodes 1 and 2 of a particular element. When we start assembling the matrices, the subscripts will be relabeled to take note of the element in question. Then 4 12 20 2 80 2 2 3 4 4 80 2 2 12 20 2 4 12 20 2 80 2 2 3 4 4 80 2 2 12 20 2 The global system of matrices equals: The load vectors are given as (12) Then each of the load vectors gives: ( ) The global load vectors are summarized as thus: Then combining the equations together, we have

GODWIN C. E. MBAH AND KINGSLEY KELECHI IBEH
The first and last rows have been covered by the boundary conditions and hence Then we solve for 2 3 4 , , u u u which gives 2 3 4 , , α α α .
The general weak solution of the Duffing equation using FEM is given as:

NUMERICAL EXPERIMENTS
In this section, we would give some numerical values to the parameters of the Duffing equation whose solution was derived in (18) above. These approximating values will make the solution be in form of a polynomial.

70
In this section, we would give some numerical values to the parameters of the Duffing equation whose solution was derived in (18) above. These approximating values will make the solution be in form of a polynomial. We take two and 0.5 ω = then, the solution (18)  , and the amount of nonlinearity β even at steady state and variation in size of  The damping has effects on earthquake occurrence as the positivity helps to reduce the tremor. Negative damping factor increases its chaotic behaviour.

GODWIN C. E. MBAH AND KINGSLEY KELECHI
In this section, we would give some numerical values to the parameters of the Duffing equation whose solution was derived in (18) above. These approximating values will make the solution be in form of a polynomial. We take two We show the numerical simulations of the behaviour of the damped Duffing oscillator which was obtained by the use of the Galerkin finite element techniques. Using MATLAB ODE 45 package, we show in figures below, the effects of even at steady state and variation in size of less than 1.
damping factor. The damping has effects on earthquake occurrence as the positivity helps to reduce the tremor. Negative

AND KINGSLEY KELECHI IBEH CONCLUSION
In this research, we have applied the use of the Finite Element Method to solving the Duffing Equation of ground motion. The FEM has been seen as a powerful tool for solving nonlinear dynamic problems which have no definite analytical solution. This method gave us a promising iterative result that such equation can be reduced to a polynomial.