Descripción de los Casos de Uso del Sistema

8.2.1 Using the Finite Element Method to Solve a Problem

Simple geometry, loading, and boundary conditions are required to achieve an exact analytical solution for a field problem. For a complicated problem that cannot be solved analytically while considering if the entire structure can be separated into small, discrete elements with simple loading and boundary conditions, one may simplify this problem as a field problem in each simple element.

FEM represents a numerical calculation approach based on the principle of virtual work:

121 8.2 Overview of the Finite Element Method

{ } { }δεε { } { }δ { } { }δ

∫

^{T σσdV =}

∫

^{u PT dV+}

∫

^{uT ΦdS (8.1)}

{ε}: strain tensor;

{σ}: stress tensor;

{u}: displacement vector;

{P}: body force in volume V;

{Φ}: surface traction on surface S.

Here δ represents a functional derivative in the calculus of variations, which is similar to a one-dimensional derivative in usual calculus. In the calculus of variations, δu is called the first variation of displacement u.

To numerically solve the problem, FEM separates a structure into discrete elements with nodes. Figures 8.1 and 8.2 show sample discrete elements in 1-D and 2-D structures.

After a structure is meshed into a number of finite elements, all of the loading functions are applied to the nodes of each element instead of on the surface or in the volume. In addition, FEM only calculates field variables on the nodes. Therefore, an interpolation function is needed to calculate values of all of the variables inside the elements. This interpolation function is associated with what we call a shape function.

For a one-dimensional linear element as shown in Figure 8.1, the shape function of a node i can be defined as:

N_i( )x = +a_i b x_i (8.2)

where: Ni( ) , ( )

, ( )

x i j

i j

j = =

≠



 1

0 (8.3)

Table 8.1. Flow chart of the basic FEM procedure 1. Problem Classification: physical phenomena analysis.

2. Mathematical Modeling: determining mathematical models, including geometries, governing equations, and appropriate solving approaches.

3. Discretization: dividing a mathematical model into a mesh of finite elements.

4. Preliminary analysis: having some analytical results, experience, or experimental results for comparison.

5. Finite element analysis:

a) Preprocessing: inputting data of geometry, material properties, boundary conditions, etc.

b) Numerical calculation: deciding interpolation functions, obtaining a matrix to describe the behavior of each element, assembling these matrices into a global matrix equation, and solving this equation to determine the results.

c) Postprocessing: listing or graphically displaying the solutions.

6. Check the results: making sure that the FEM simulation has been carried out correctly and then comparing the FEM results to the preliminary analysis.

If the two nodes of the element are located at x₁ and x₂, then:

Solving Equation (8.4) gives:

a x

Therefore, the shape functions become:

N

For a 2-D linear triangular element shown in Figure 8.3, the shape function of node i can be defined as:

N_i( , )x y = +ai b x c yi + i (8.7)

Element 1 Element 2

F₁ F₂

Figure 8.1. Sample one-dimensional problem: the structure is disintegrated to two bar elements.

F₁ F₂ F₃ Fs

Element Node

Figure 8.2. Sample two-dimensional problem: the structure is disintegrated to sixteen triangular elements.

123 8.2 Overview of the Finite Element Method

Substituting Equation (8.7) into Equation (8.8) yields:

a x y x y

where Δ is the element area.

Shape functions for the 3-D elements can be derived similarly.

After the shape function is determined for an element, one can obtain the displacement distribution in the element. For the linear triangular element in Figure 8.3, the displacements in the x and y directions are:

u

Based on elasticity theory, strains in the element are defined as:

ε

Figure 8.3. Sample two-dimensional linear triangular element. ui and vi represent displacements of node i in x and y directions. Node is located at (xi, yi).

Substituting Equation (8.10b) into Equation (8.11) yields:

In addition, the constitutive equation in the element can be written as:

σσ εε

{ }

=

[ ]{ }

^{D =}

[ ][ ]{ }

^{D B d ( )e (8.13)}

For a plane strain problem in an isotropic material, [D] becomes:

[ ]D =

where E and ν indicate Young’s modulus and Poisson’s ratio.

For a plane stress problem in an isotropic material, [D] becomes:

[ ] ( )

For an axisymmetric 2-D problem in cylindrical coordinates in an isotropic material, [D] becomes:

Substituting Equations (8.10b), (8.12), and (8.13) into Equation (8.1) yields:

δ d B D B d N P N

{ }

^T

( ∫

^{[ ] [ ][ ]T dV{ }( )e −}

( ∫

^{[ ] { }T dV+}

∫

^{[ ] { }T ΦdS}

) )

^{=0 (8.17)}

Let:

[ ]k^{( )e} =

∫

[ ] [ ][ ]B D B^T dV ^and { }F^{( )e} =

( ∫

[ ] { }N P^T dV+

∫

[ ] { }N^{T Φ}dS

)

125 8.2 Overview of the Finite Element Method

then Equation (8.17) becomes:

[ ] { }k^{( )e} d^{( )e} ={ }F^{( )e} (8.18) where [ ]k^{( )e} is called the element stiffness matrix and { }F^{( )e} is the equivalent element load vector. Equation (8.18) is the governing equation for a single element.

If there are n nodes in the entire structure, by assembling all of the elements in the entire structure, the governing equation becomes:

[ ]K₂n×₂n{ }d₂n×₁={ }F ₂n×₁ (8.19) where:

the Nodal displacement vector is as follows:

{ } [d = u v ... ui vi ... uj vj ... uk vk ... un vn]T

1 1 (8.20)

the Equivalent load vector is as follows:

{ } [F = f_x,1 f_y,1 ... f_{x i,} f_{y i,} ... f_{x j,} f_{y j,} ... f_{x k,} f_{j k,} ... f_{x n,} f_{y n,} ]^T (8.21) the Global stiffness matrix is as follows:

[ ]

For plane strain problems, the element in stiffness matrix [K] is:

K Eh b b c c b c c b

8.2.2 Quadratic Element

In FEM, linear elements are simple and easily calculated. However, because the shape functions of these elements are linear, the displacement distributions are also linear. In other words, the deformation of a linear element is restricted. For instance,

a deformed linear triangular element will always be a triangle without any curved edges. In addition, according to Equation (8.12), the strain in a linear element is a constant. Because a real structure usually does not have these restrictions, it requires that FEM uses many more elements to achieve accurate solutions for structures with complex geometries and/or energy distributions.

An efficient approach to reduce the required element number is to employ nonlinear elements. Figure 8.4 illustrates a quadratic triangular six-node element.

Besides three vertex nodes, i, j, k, three side nodes, l, m, and s, are added to the element. The shape function of this quadratic triangular element is:

Ni( , )x y = +ai b x c y d xi + i + i ²+e xy f yi + i ² (8.24) For structures with simple geometries but complex energy distributions, bilinear elements will be a good choice. The shape function of a bilinear element is:

Ni( , )x y = +ai b x c y d xyi + i + i (8.25)

8.2.3 Dynamic Problem

Wave propagation is a dynamic problem. The governing equation of FEM for a dynamic problem is:

M d C d K d F

[ ] { }

^{ +}

^{[ ]} { }

^{ +}

^{[ ] { }}

⁼

^{{ }}

^(8.26)

which can be considered as generalized Newton’s second law. Here M

[ ]

^{and C}

[ ]

represent the mass matrix and damping matrix while d

{ }

^=d

^{{ }}

d ^/dt is the velocity vector and d

{ }

^=d2

^{{ }}

d ^/dt2 is the acceleration vector.

If a 2-D element has a thickness h and density ρ, the mass matrix becomes:

M N N

[ ]

^=ρh

∫ ∫ [ ] [ ]

^{T dxdy (8.27)}

x

y vj

v_m

vk

v_l

vi

v_s u_s u_k

u_l u_m

u_i (x_j , y_j)

(x_m , y_m)

(x_i , y_i)

(x_k , y_k)

(x_s , y_s) (x_l , y_l)

uj

Figure 8.4. Sample two-dimensional quadratic triangular element. ui and vi represent displacements of node i in the x and y directions. Node i is located at (x_i, y_i).

127 8.2 Overview of the Finite Element Method

For Rayleigh damping, the damping matrix is defined as a linear combination of the mass matrix and the stiffness matrix:

C M K

[ ]

^=α

[ ]

^+β

[ ]

(8.28)

The coefficients α and β can be determined by experiments. It has been shown that α has very little influence on high-frequency problems. Because ultrasonic waves are of high frequency, the governing equation becomes:

M d K d K d F

[ ] { }

^{ +β}

^{[ ]} { }

^{ +}

^{[ ] { }}

⁼

^{{ }}

^(8.26)

For a harmonic wave with circular frequency ω in an undamped material, Equation (8.26) can be simplified as:

K M d F

[ ]

⁻

[ ]

(

^ω2

) { }

⁼

{ }

_(8.29)

Matrix K

[ ]

^−ω2

[ ]

M is called the dynamic stiffness matrix. Based on Equation (8.29), the problem can be solved in the frequency domain instead of the time domain.

To solve a dynamic problem in the time domain, numerical calculation methods in the time domain are needed. A direct integration method is the most popular numerical approach for calculating the time domain response. The numerical calculation method is a finite difference method that obtains derivation of time t from Taylor’s polynomial. The most widely used direct integration methods include the center difference method, the Newmark method, and the Wilson θ method. We first discuss the classic center difference method here.

In the classic center difference method, if the time increment is Δt, the velocity and acceleration at time step n can be approximated as:

{ }dn { }dn ({ }dn { } )dn

substituting Equations (8.30)~(8.33) into Equation (8.26) provides:

1 1 When using the direct integration method, the initial displacement, velocity, and acceleration of the structure must be known to solve the problem.

The classic center difference method is a simple approach, although it is not unconditionally stable. To optimize the stability of the direct integration method, one may employ the Newmark method or the Wilson θ method. Because the Wilson θ method has large “damping” for high-frequency problems, the Newmark method is a better choice for ultrasonic wave simulation.

The Newmark method approximates the velocity and displacement at time step n as:

   

d d d d

{ }

n+ ⁼

{ }

n⁺ t

( ⁽

⁻

⁾ { }

n⁺

{ }

n+

)

^{≤ ≤}

1 Δ 1 γ γ 1 (0 γ 1) (8.35)

d d d d d

{ }

n^{+ =}

{ }

n⁺ t

{ }

n⁺ t

( (

⁻

) { }

n⁺

{ }

n+

)

^{≤ ≤}

1

2

2 1 2 2 1 0 2 1

Δ Δ β  β  ( β ) (8.36)

When 0 5. ≤ ≤γ 1 and 0 25. ≤ ≤β 0 5. , the Newmark method is unconditionally stable.

8.2.4 Error Control

FEM is a numerical calculation method that produces results that do not always match the exact solutions, so it is important to reduce the errors in FEM application and to make the FEM solutions converge to the exact solutions. Because most FEM simulations of ultrasonic wave behavior are accomplished by commercial FEM software, in this section, we discuss how to reduce common errors occurring when using FEM software.

Several types of errors must be considered when using FEM software:

1. Incorrect mathematical model. If the physical phenomenon of a problem

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