Lifetime-Oriented Structural Design Concepts- P15 doc

380 4 Methodological Implementation◦ selective reduced integration [397, 649] and hourglass control [283] ◦ assumed natural strain concept [398, 247] ◦ B-bar methods [850, 789] ◦ enhance

378 4 Methodological Implementation

( •) represents all state variables ˙ uNF, uNF and spatial gradients ∇ uNF.

On the basis of equation (4.14) the weak form of the coupled multiphysics

system δW is generated in equation (4.15) by the weighted summation of the individual weak forms δWf, whereby Af is introduced to adapt physical units and dimensions of coupled fields.

4.2.3.2 Linearized Weak Form of Coupled Balance Equations

In order to prepare the weak form for the numerical solution with the Newton-Raphson scheme, the weak form is expanded in a Taylor series about the trial solution of all state variables

The increment of weak forms is generated by summation of individual

portions in terms of the increments of field variables Δug and gradients of field variables ∇Δ ug with g ∈ [1, NF].

It is worth to mention that the derivative with respect to gradient

∇ u is performed explicitly in oder to obtain an advantageous format for

the finite element discretization discussed in Section 4.2.4.2 The individual terms in equation (4.18) are expressed as follows:

4.2.4 Spatial Discretization Methods

Authored by Detlef Kuhl and Christian Becker

Within the framework of the semdiscretization technique applied to solve durability single- and multiphysics problems, the spatial discretization is re- alized by the finite element method (see e.g [90, 106, 223, 224, 870, 871]) The scientific and industrial oriented literature documents the broad range of applications of this method for the spatial discretization of differential equa- tions Highly non-linear problems, stationary and transient problems as well

as single- and multifield problems can by solved adopting the finite element method.

implementa-the element size (h-method) and increasing implementa-the polynomial degree of ansatz functions (p-method), respectively Furthermore, special element techniques

can be used to overcome the well known locking problem In summary, two main philosophies are used in the context of structural mechanics, to obtain high quality numerical results applying the finite element method.

• Low order finite element methods combined with methods to prevent

lock-ing [637, 790]:

380 4 Methodological Implementation

◦ selective reduced integration [397, 649] and hourglass control [283]

◦ assumed natural strain concept [398, 247]

◦ B-bar methods [850, 789]

◦ enhanced assumed strain concept [747, 742]

• Higher order finite element methods using different kind of higher order

ansatz polynomials:

◦ multidimensional Lagrange polynomials [870, 452]

◦ Legendre based hierarchical one-, two- and three-dimensional shape

functions [72, 717, 246]

A literature review about high quality, low order finite element methods makes clear that the related element techniques are separately developed for selected applications in structural mechanics But computational durability mechanics

is characterized by manifold various underlying differential equations Therfore, the more general higher order finite element method is presented in the following.

4.2.4.2 Generalized Finite Element Discretization of Multifield

Problems

The numerical analysis of non-linear multiphysics problems can be subdivided

in the spatial finite element discretization, the temporal discretization and the iterative solution of the resulting non-linear algebraic equation In the present

section a detailed description of the spatial p-finite element discretization of

generalized multiphysics problems is given.

4.2.4.2.1 Approximations

The finite element formulation of the original and the linearized weak forms of multiphysics problems is based on the approximation of test functions, state variables and their gradients by shape functions and nodal values of state

variables Therefore, ND-dimensional anisotropic shape functions of arbitrary polynomial degrees pd for the ND spatial directions d are designed based on

one-dimensional Lagrange shape functions Furthermore, the approximation

of state variables is given and the transformation between natural and physical element coordinates is performed by the Jacobi transformation.

One-dimensional Lagrange shape functions of polynomial degrees pdcan

be generated for every spatial direction d ∈[1, ND] by the product

d with i, k ∈ [1, pd + 1] Consequently, the derivatives of shape

functions are also calculated for arbitrary polynomial degrees pd.

k=1

k=i k=l

ξd k− ξd

ξd k− ξi d

(4.21)

ND-dimensional isotropic or anisotropic shape functions are generated by the

product of one-dimensional shape functions

Shape functions (4.22) and and spatial derivatives of shape functions (4.23) are applied for the approximation of state variables

shape function N11 derivative N11

Fig 4.4 Illustration of isotropic Lagrange shape functions and derivatives of

shape functions by means of a cubic planar finite element

Fig 4.5 Illustration of anisotropic Lagrange shape functions and derivatives of

Lagrangeshape functions by means of a cubic-linear planar finite element

Table 4.1 Multi-dimensional Lagrange shape functions and specialization to one-,

two- and three-dimensional finite elements

element type position vector

X

field variable

u

integral

−1 • dξ1A

  !

dV ξ

1

−1

1

−1 • dξ1dξ2H

  !

dV ξ

1

−1

1

−1

1

in terms of nodal state variables uei

f and ˙ uei f Furthermore, equations (4.24)

and (4.25) are used for the approximation of δuf, Δuf, δ ∇ uf and Δ ∇ uf

4.2.4.2.2 Non-Linear Semidiscrete Balance

Inserting the approximations discussed above into the weak form of

multi-physics problems (4.14) and (4.15) for individual finite elements e ∈ [1, NE]

yields the discretized weak form on the element level

f according to the nodal values of the test

function δuei f.

Niφ f dA

(4.31)

After assembling the element quantities over element nodes, elements and tensor fields, application of the fundamental lemma of variational cal- culus and consideration of initial values the semidiscrete initial value problem

ri(¨ u , ˙ u , u ) = r u (t0) = u0, u ˙0, ¨ u0 (4.32)

is obtained In the non-linear second order vector differential equation (4.32) the vectors ri, r , u , ˙ u and ¨ u represent the generalized vectors of internal forces, external forces, primary variables, first and second temporal rates of primary variables, respectively, whereby every structural vector

contains the nodal values of all contributing fields f As particular example

the assembling of the generalized vector of internal forces ri based on the nodal element internal force tensors rei

The application of the present generalized multiphysics finite element cept for the discretization of the chemo-mechanical damage model yields the element tensors of internal and external forces (rei

4.2.4.2.3 Linearized Semidiscrete Balance

Applying the approximation procedure to the linearized weak form (4.17-4.19)

on the element level yields its discrete counterpart

in terms of generalized tangent stiffness tensors keij f g( ˙ ue NF, ue

NF) and generalized tangent damping tensors deij f g( ˙ ue NF, ue NF) according to the test

Because of the time independent balance of momentum (4.7) the tensors d11eij,

deij12 and deij21 vanish In equations (4.40) and (4.42) the following abbreviations are used:

4.2.4.2.4 Generation of Element and Structural Quantities

The integrals in equations (4.31) and (4.36) are computed by the Legendre quadrature with NG = ND

Gauß-d=1NGd integration points ξl d

d, ld ∈ [1, NGd] contained in vectors ξl and the weights αl based on the one- dimensional integration rule, see e.g [870].

Figure 4.6 illustrates the final calculation of element quantities of generalized

multiphysics p-finite elements It is obvious that specific multiphysics problems

as described in Chapter 3 can be implemented on the model or Gauß point level marked within the algorithmic set-up Internal variables κf are just managed on the finite element level Manipulations of internal variables take exclusively part on the material point level.

ND loops over Gauss points l d ∈ [1, NG d]

next Gauß point l d

loop over element nodes i ∈ [1, NN]

coordinates and weight of Gauß point ξ = [ξ l1 ξ l2ξ l3]T , α = α l1α l2α l3

shape functions and natural derivatives N i(ξ ), ∇ ξ N i(ξ)

Jacobideterminant and inverse Jacobi tensor | J(ξ)|, J −1(ξ)

state variables X(ξ),u f(ξ), ˙u f(ξ)physical gradient of shape functions ∇N i(ξ) =J −T(ξ)·∇ ξ N i(ξ)

model level Θ˙f,Φ f ,∂ ˙ Θ f /∂ ∇ u g ,∂ ˙ Θ f /∂ u g ,∂ ˙ Θ f /∂ ˙ u g ,∂ Φ f /∂ ∇ u g ,

summation and assembly α | J | r ei f →r, α | J | r ei →ri

generalized tangent damping tensor d eij f g

generalized tangent stiffness tensor k eij f g

summation and assembly α | J | d eij f g →D, α | J | k eij f g →K

Fig 4.6 Computation of generalized element tensors of external and internal forces

and generalized tangent damping and stiffness tensors of multiphysics p-finite

ele-ments

4.2.4.3 p-Finite Element Method

The p-finite element method (p-FEM) is the exact counterpart to the h-finite element method (h-FEM) Whereas in the h-FEM a mesh of low-order

elements is refined by increasing the number of elements that are naturally

388 4 Methodological Implementation

h-mesh p-mesh

0-1-2-3-4-5-6

L = 1, 0

p(X1) = sin(−8 X1)

X1

Fig 4.7 Sinusoidal loading of a truss member (left), Rel error of internal energy

plotted over the number of degrees of freedom (right)

smaller than the original ones, the p-FEM approach is similiar to the

well-known Ritz-method The basic idea of the approach is to use a relative coarse finite element mesh If the quality of the numerical solution has to

be improved the mesh remains unchanged but the polynomial degree of the shape functions, and therefore the polynomial degree of the approximation,

is increased subsequently In the limit case of p → ∞ the exact solution is

gained.

A considerable advantage of the p-finite element method in the case of

a smooth solution is the exponential convergency compared to the almost linear convergence rate for the h-method This is illustrated by an example

that was analyzed in [245] and deals with a truss member that is loaded by

a sinusoidal force that is acting along the member’s length (see Figure 4.7) The internal energy and the relativ error are defined as:

Basically, there are two concepts of higher order shape function concepts

within the p-FEM: the non-hierarchical Lagrange concept and the

hierar-chical concept based on the Legendre polynomials.

4.2.4.3.1 Onedimensional Higher-Order Shape Function Concepts

Before dealing with details on threedimensional shape function concepts the basic concept of higher order approximations is illustrated in the onedimen- sional case The generation of shape functions of the Lagrange-Type has already been introduced in Section 4.2.4.2.1.

4.2.4.3.1.1 Shape Functions of the Legendre-Type

Legendre polynomials are the solutions Pn(ξ) of the homonymous

chosing P0(ξ) = 1 and P1(ξ) = ξ as starting polynomials Before the

poly-nomials (4.47) are applied within a shape function concept, they are fied such that their value vanish at the outer vertices of an onedimensional element:

modi-Φi(ξ) =  1

2 (2 i − 1) [Pi(ξ) − Pi−2(ξ)] , i ≥ 2. (4.48)

Figure 4.8 shows a selection of some modified higher-order Legendre polynomials Within an onedimensional hierarchical shape function concept the first two shape functions are the standard linear Lagrange shape functions

0.2

0 -0.2

-0.4

-0.6

Fig 4.8 Modified Legendre-polynomials for polynomial degreesp = 2, 3, 4, , 7

4.2.4.3.1.2 Comparison of Both Shape Function Concepts

In general the numerical solutions of both concepts are identical if the same approximation degree is used This is because hierarchical and non-

hierarchical approximation describe the same polynomial Lp of degree p

Lp(ξ) = a0+ a1ξ1+ a2ξ2+ + apξp. (4.51) Solely the meaning of the discrete degrees of freedom differ from each other This topic will be dealt with in a subsequent paragraph Figure 4.9 shows the sets of shape functions for the Lagrange and for the Legendre concept respectively for the approximations p = 1, 2, 3 As it can be seen easily,

the shape functions of the Lagrange type fullfill the interpolation property

be newly generated when the approximation order is increased Regarding the hierarchical concept it can be seen that an increase of the approximation order

from p to p + 1 results in generating only one new shape function of degree p+1 whereas the existing shape functions remain unchanged This is the main

characteristic of the hierarchical concept Figure 4.10 illustrates the structure

of the element stiffness matrices K and of the internal load vector r of both concepts for different polynomial degrees within an arbitrary linear problem.

It can be seen that the bandwith of the stiffness matrix is smaller in the Legendre concept Furthermore, the hierarchical structure of the matrices and vectors within the Legendre concept is obvious In linear problems the

10.50-0.5

-1

10.50-0.5

-1

10.50-0.5

Fig 4.9 Set of hierarchically organized shape functions of the Legendre type for

polynomial degrees p = 1, 2, 3 (left), Shape functions of the Lagrange type holding for the interpolation property for polynomial degrees p = 1, 2, 3 (right)

hierarchical concept can be used in adaptive p-refinements where the quality

of the approximation is controlled by an error-estimator or indicator If a

p-refinement is needed most parts of the existing matrices and vectors can be

used and only few entries have to be newly calculated Furthermore it can

be seen that within an onedimensional hierarchical concept the higher order degrees of freedom are decoupled from the other ones, because there are only entries on the main diagonal of the element stiffness matrix In general, in a hierarchical concept the resulting equation system has a better conditioning [245].

The meaning of the nodal degrees of freedom within both concepts is as follows: In the Lagrange concept the nodal values represent the value of the

392 4 Methodological Implementation

Legendrediscretization Lagrangediscretization

p=3 p=2

Fig 4.10 Comparison of the structure of element vectors and matrices for the

Legendre- and Lagrange-concept for polyomial degrees p = 1, 2, 3

field variable at the corresponding node of the element Because of the lack of the interpolation property the nodal values of the nodes within the Legendre concept - except the values at the vertices of a brick element - can not be interpreted that way The higher-order degrees of freedom represent

- multiplied with the corresponding shape function - a p-fraction of the

resulting distribution of the solution.

4.2.4.3.2 3D-p-Finite Element Method Based on Hierarchical Legendre

Polynomials

After illustrating the main properties of the onedimensional hierarchical shape function concept in comparison to the Lagrange concept, the approximation techniques of the threedimensional hierarchical shape function concept are described subsequently The basis of the threedimensional implementation

is the brick element illustrated in Figure 4.11 Therein the definition and numbering of element vertices, edges and faces is given.

4.2.4.3.2.1 Generation of 3D-p-Shape Functions

The threedimensional shape functions result from a spatial mutliplication

of the onedimensional ones (4.49,4.50), leading to:

contains the nodal values of all contributing fields f As particular... (4.42) the following abbreviations are used:

4.2.4.2.4 Generation of Element and Structural Quantities

The integrals in equations (4.31) and (4.36) are computed