MultiPhysicsVault/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_003.md

1.4 References

1. ZIENKIEWICZ, O. C. The Finite Element Method, McGraw-Hill, 1977.
2. ODEN, J. T., Finite Elements of Nonlinear Continua, McGraw-Hill, 1972.
3. DESAI, C. S. and ABEL, J. F., An Introduction to the Finite Element Method, Van Nostrand Reinhold, New York, 1972.
4. GALLAGHER, R. H., Finite Element Analysis—Fundamentals, Prentice Hall, 1975.
5. HINTON, E. and OWEN, D. R. J., Finite Element Programming, Academic Press, 1977.
6. HINTON, E. and OWEN, D. R. J., An Introduction to Finite Element Computations, Pineridge Press, Swansea, U.K., 1979.

Chapter 2 One-dimensional nonlinear problems

2.1 Introduction

Several classes of nonlinear problems of interest in many branches of science and engineering can be reduced to the solution of a system of simultaneous equations in which the equation coefficients are dependent on some function of the prime variables. ^{(1)} In this chapter some basic techniques for the numerical solution of such problems are examined. In order to introduce the essential details of the solution processes as simply as possible, the applications will be restricted to one-dimensional situations. In particular, elasto-plasticity, nonlinear elasticity problems and systems governed by a nonlinear quasi-harmonic equation will be considered. In each case a computer program will be developed and its use illustrated by application to simple problems. The aim of this chapter is to prepare the reader for the more comprehensive two-dimensional treatment of these topics which will be undertaken in Chapters 6–9. Indeed, all the essential features of nonlinear finite element analysis detailed in these later chapters will be recognisable from the simple treatment considered here. It should be emphasised that the subroutines developed in this chapter will not be used in the main finite element programs discussed in Parts II and III.

2.2 Basic numerical solution processes for nonlinear problems

The use of finite element discretisation in a large class of nonlinear problems results in a system of simultaneous equations of the form

\boldsymbol {H} \boldsymbol {\varphi} + \boldsymbol {f} = 0, \tag {2.1}

in which \varphi is the vector of the basic unknowns, f is the vector of applied ‘loads’ and H is the assembled ‘stiffness’ matrix. For structural applications, the terms ‘load’ and ‘stiffness’ are directly applicable, but for other situations the interpretation of these quantities varies according to the physical problem under consideration.

If the coefficients of the matrix H depend on the unknowns \varphi or their derivatives, the problem clearly becomes nonlinear. In this case, direct solution of equation system (2.1) is generally impossible and an iterative scheme must be adopted. Many options remain open for the iterative

sequence to be employed. Some of the most generally applicable methods available will now be outlined.

2.2.1 Method of direct iteration (or successive approximations)

In this approach ^{(2)} successive solutions are performed, in each of which the previous solution for the unknowns \varphi is used to predict the current values of the coefficient matrix H(\varphi) . Rewriting (2.1) as

\varphi = - [ H (\varphi) ] ^ {- 1} f, \tag {2.2}

then the iterative process yields the (r + 1)^{\mathrm{th}} approximation to be

\varphi^ {r + 1} = - [ H (\varphi^ {r}) ] ^ {- 1} f. \tag {2.3}

If the process is convergent then in the limit as r tends to infinity \varphi^r tends to the true solution.

It is seen from (2.3) that it is necessary to recalculate the 'stiffness' matrix H for each iteration. To commence the process, an initial guess for the unknown \varphi is required in order to calculate H . Generally a value of \varphi^0 based on the solution for an average material property throughout the region is found to be satisfactory. If the nonlinearity of the material properties is very marked at certain values of \varphi , an approximate prescription of the field variable at all nodes may be necessary.

For practical purposes, the iterative process is deemed to have converged when some measure (usually a norm of the nodal unknowns) of the change in the unknown \varphi between successive iterations has become tolerably small. The process is illustrated diagrammatically for a single variable in Figs 2.1 and 2.2, in which case the matrix H and vector \varphi reduce to the scalar equivalents H and \phi . The assumed dependence of H on \phi is a basic problem function which must be prescribed before solution can commence. This material property is included in Figs 2.1 and 2.2 and, for convenience, the relationship between H(\phi). \phi and \phi is prescribed rather than the H(\phi) - \phi dependence. Figure 2.1 shows the convergence paths for initial trial values, \phi^0 , which are below and above the true solution, \phi_T , and for a convex H - \phi relation. From the initial trial value, \phi^0 , the corresponding value of H is immediately given from the prescribed H(\phi). \phi - \phi relationship, to be H^0 . Equation (2.3) is then solved to give \phi^1 . The value of H corresponding to \phi^1 is then determined from the H(\phi). \phi - \phi relationship and (2.3) then resolved to obtain \phi^2 . This cycling process is continued until \phi^{n-1} and \phi^n are deemed to be sufficiently close, indicating that convergence has occurred. The quantity H^r is represented by the slope of the secant to the H - \phi curve and decreases with increasing values of \phi . Both the high and low initial trial solutions produce monotonic convergence paths. Figure 2.2 shows the unsuitability of the method for problems with a concave H - \phi relationship. Both low and high initial trial solutions produce convergence paths which oscillate around the true solution. Although the solution converges for the

line

Basic variable, φ	H(φ)φ (Slope H⁰)	H(φ)φ (Slope H¹)	H(φ)φ (Slope H²)
φ⁰	~0.0	~0.0	~0.0
φ¹	~0.5	~0.3	~0.1
φ²	~1.0	~0.8	~0.3
φ³	~1.5	~1.2	~0.6

line

x	φ
0	0
1	0.5
2	0.75
3	0.9
4	0.95
5	0.98

(a) Low initial solution

text_image

H(φ)φ f H^1 H^2 Slope H^0 φ^3 φ^2 φ^1 φ^0

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x	φ
0	0
1	φ
2	φ
3	φ
4	φ
5	φ

(b) High initial solution
Fig. 2.1 Direct iteration method for a single variable problem—convex H-\phi relation.

single variable case, in multi-degree of freedom problems the coupling of stiffness terms is likely to lead to instability of the iterative process. A disadvantage of the direct iteration method is that convergence of the solution scheme is not guaranteed and cannot be predicted at the initial solution stage.

2.2.2 The Newton-Raphson method

During any step of an iterative process of solution, (2.1) will not be satisfied unless convergence has occurred. A system of residual forces can be assumed

line

φ	H(φ)φ (H¹)	H(φ)φ (H²)	Slope H⁰
0	0	0	0
φ⁰	~0.5	~0.2	~0.1
φ²	~1.0	~0.5	~0.2
φ³	~1.5	~0.8	~0.3
φ¹	~2.0	~1.0	~0.4

line

x	φ
0	0
1	1
2	0.5
3	0.75
4	0.5
5	0.75

(a) Low initial solution

line

φ	H(φ)φ (H²)	H(φ)φ (H¹)
0	0	0
φ¹	H²	H¹
φ³	H²	H¹
φ²	H²	H¹
φ⁰	H²	H¹

line

x	φ
0	φ
1	φ_T
2	φ_T
3	φ_T
4	φ_T
5	φ_T

(b) High initial solution
Fig. 2.2 Direct iteration method for a single variable problem—concave H-\phi relation.

to exist, so that

\psi = H \varphi + f \neq 0. \tag {2.4}

These residual forces \psi can be interpreted as a measure of the departure of (2.1) from equilibrium. Since H is a function of \varphi and possibly its derivatives, then at any stage of the process, \psi = \psi(\varphi) .

If the true solution to the problem exists at \varphi^r + \Delta \varphi^r then the Newton-Raphson approximation ^{(2)} for the general term of the residual force vector, \psi^r corresponding to solution at \varphi^r is

\psi_ {i} ^ {r} = - \sum_ {j = 1} ^ {N} \Delta \phi_ {j} ^ {r} \left(\frac {\partial \psi_ {i}}{\partial \phi_ {j}}\right) ^ {r}, \tag {2.5}

in which N is the total number of variables in the system and the superscript r denotes the r^{th} approximation to the true solution. Substituting for \psi_{i} from (2.4), the complete expression for all the residual components can be written in matrix form as

\psi (\varphi^ {r}) = - J (\varphi^ {r}) \Delta \varphi^ {r}. \tag {2.6}

in which a typical term of the Jacobian matrix J is

J _ {i j} = \left(\frac {\partial \psi_ {i}}{\partial \phi_ {j}}\right) ^ {r} = h _ {i j} ^ {r} + \sum_ {k = 1} ^ {m} \left(\frac {\partial h _ {i k}}{\partial \phi_ {j}}\right) ^ {r} \phi_ {k} ^ {r}, \tag {2.7}

where h_{ij} is the general term of matrix H. The last term in (2.7) gives rise to nonsymmetric terms in the Jacobian matrix. If these nonsymmetric terms are neglected in order to maintain symmetry, then substitution of (2.7) in (2.6) results in

\boldsymbol {H} \left(\varphi^ {r}\right). \Delta \varphi^ {r} = - \psi \left(\varphi^ {r}\right). \tag {2.8}

Or since

\Delta \varphi^ {r} = \varphi^ {r + 1} - \varphi^ {r}, \tag {2.9}

equation (2.8) reduces, on use of (2.4), to

\boldsymbol {H} \left(\boldsymbol {\varphi} ^ {r}\right). \boldsymbol {\varphi} ^ {r + 1} + \boldsymbol {f} = 0. \tag {2.10}

This equation is identical to equation (2.3), Section 2.2.1, which governs the method of direct iteration. Therefore in order to achieve the better convergence rate associated with the Newton–Raphson process it is essential that the unsymmetric terms in J be retained.

The explicit form of the nonlinear terms in (2.7) will clearly depend on the way in which the stiffness matrix coefficients, h_{ij} , depend on the unknowns, \varphi . The terms of the Jacobian matrix, given in (2.7), can be assembled to give the general expression

\boldsymbol {J} (\varphi) = \boldsymbol {H} (\varphi) + \boldsymbol {H} ^ {\prime} (\varphi), \tag {2.11}

where the last term contains the unsymmetric terms only. The Newton-Raphson process can be finally written, using (2.6) and (2.11), in the form

\Delta \varphi^ {r} = - [ J (\varphi^ {r}) ] ^ {- 1}. \psi (\varphi^ {r}) = - [ H (\varphi^ {r}) + H ^ {\prime} (\varphi^ {r}) ] ^ {- 1} \psi (\varphi^ {r}). \tag {2.12}

This allows the correction to the vector of unknowns \varphi to be obtained from the residual force vector \psi for any iteration. Again an iterative approach must be followed, with the vector of unknowns \varphi being corrected at each stage according to (2.12) until convergence of the process is deemed to have occurred. The technique is illustrated schematically in Figs 2.3 and 2.4 for

line

φ	H(φ)φ (Slope J(φ⁰))	H(φ)φ (J(φ¹))	H(φ)φ (ψ⁰)	H(φ)φ (Δφ⁰)	H(φ)φ (Δφ¹)
φ⁰	Low	Low	Low	Low	Low
φ¹	High	High	High	High	High
φ²	High	High	High	High	High

line

φ_T	φ
0	0.0
1	0.2
2	0.3
3	0.4
4	0.5
5	0.5

(a) Low initial solution

line

f	H(φ)φ (Slope J(φ⁰))	Δφ⁰ (Slope J(φ⁰))	ψ¹ (Slope J(φ⁰))	Δφ¹ (Slope J(φ⁰))
0	0	0	0	0
φ¹	~0.5	~0.5	0.5	0.5
φ²	~1.0	~1.0	1.0	1.0
φ⁰	~1.5	~1.5	1.5	1.5

line

x	φ	φ_T
0	1.0	0.0
1	0.2	0.0
2	0.3	0.0
3	0.4	0.0
4	0.5	0.0
5	0.6	0.0

(b) High initial solution
Fig. 2.3 The Newton-Raphson method for a single variable problem—convex H-\phi relation.

Fig. 2.4 The Newton–Raphson method for a single variable problem—concave H-\phi relation.

a single variable situation. Solution to the nonlinear problem will be achieved when the residual force \psi vanishes, since this term directly measures the lack of equilibrium of the governing equation as indicated in (2.4). A trial value \varphi^{0} of the basic unknown is assumed and the material stiffness associated with this value calculated according to the prescribed H-\varphi relationship.

The residual force, \psi^0 is then calculated from (2.4) and the Jacobian evaluated according to (2.7). The correction \Delta \varphi^0 to the first approximation for the basic unknown, can finally be found from (2.12). Thus an improved approximation to the solution has been found, as \varphi^1 = \varphi^0 + \Delta \varphi^0 . This process can then be continually repeated until the residual force, \psi^n , is sufficiently small; or equivalently that \varphi^{r-1} and \varphi^r are sufficiently close. The Newton-Raphson process generally gives a more rapid and stable convergence path than the direct iteration method.

2.2.3 The tangential stiffness method

For structural applications the matrix H can be interpreted physically as the stiffness matrix of the structure. For nonlinear situations, in which the stiffness depends on the degree of displacement in some manner, H is equal to the local gradient of the force/displacement relationship of the structure at any point and is termed the tangential stiffness. The analysis of such problems must proceed in an incremental manner since the solution at any stage may not only depend on the current displacements of the structure, but also on the previous loading history. Consequently the problem can be linearised over any increment of load and therefore the matrix, which contains the nonlinear terms, can be discarded from (2.11) and (2.12). With this modification, the solution process is identical to that described in the previous section and for this reason the method is sometimes termed a generalised Newton–Raphson method.

The solution algorithm is illustrated in Fig. 2.5; again for a single variable situation. Solution is commenced from a trial value \varphi^{0} of the unknown (for structural problems the starting position of solution is almost invariably \varphi^{0} = 0 ). The tangential stiffness, H(\varphi^{0}) , corresponding to this displacement state is then determined and the residual force \psi^{0} calculated according to (2.4). The correction, \Delta\varphi^{0} , to the trial value is computed according to the linearised form of (2.12), which is

\Delta \varphi^ {r} = - [ H (\varphi^ {r}) ] ^ {- 1}. \psi (\varphi^ {r}) \tag {2.13}

An improved approximation to the unknown is then obtained as \varphi^{1} = \varphi^{0} + \Delta\varphi^{0} . This iterative process is then continued until the solution converges to the nonlinear solution which is indicated by the condition that \psi^{r} practically vanishes.

2.2.4 The initial stiffness method

In the methods described in the three previous sections, the complete factorisation (or reduction) and solution of the full set of simultaneous equations describing the discretised structure is essential for each iteration. For the method of direct iteration the equation solution indicated by (2.3) is necessary, whilst the Newton–Raphson technique and tangential stiffness method demand the equation solutions indicated by (2.12) and (2.13)