MultiPhysicsVault/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_051.md

We have adopted standard and modified Newton methods throughout this text as they are the most widely used approaches. Though they work well they do have certain disadvantages. The initial stiffness method is slow to converge in cases in which there is a high degree of nonlinearity. The modified Newton methods provide better convergence properties but they diverge during elastic unloading and they can lead to ill-conditioned or singular Jacobian matrices K near the limit load.

Newton methods are sometimes employed with a slight modification during an iteration in which

\boldsymbol {K} ^ {(i)} \Delta \boldsymbol {d} ^ {(i)} = \psi^ {(i)}, \tag {12.20}

and in which the new displacement vector is given as

\boldsymbol {d} ^ {(i + 1)} = \boldsymbol {d} ^ {(i)} + a ^ {(i)} \Delta \boldsymbol {d} ^ {(i)}, \tag {12.21}

where we could take a^{(i)} as much less than 1 for safety or more than 1 for more rapid convergence. Nayak ^{(88)} introduced an acceleration technique in which a^{(i)} is replaced by a diagonal matrix. Basu ^{(89)} later simplified this technique.

Although the modified Newton methods with fixed values of \alpha^{(t)} is employed by certain analysts, it has been suggested ^{(90)} that we should reject it in favour of a modified Newton with a line search which involves finding a value of \alpha^{(t)} which minimises the total potential energy \pi(\boldsymbol{d}^{(t+1)}) or the value of

Q = \left| \left[ \boldsymbol {d} ^ {(i)} \right] ^ {T} \psi \left(\boldsymbol {d} ^ {(i + 1)}\right) \right|. \tag {12.22}

12.5.2 Quasi-Newton method

Over the past twenty years there has been a rapid development of computer-oriented, sequential search methods in the fields of optimisation and mathematical programming. Of these techniques, the variable metric (Quasi-Newton) method and the method of conjugate gradients show the greatest potential in nonlinear finite element analysis.

The Quasi–Newton method was introduced to finite element computations by Matthies and Strang. ^{(91)} The main idea is to update the matrix K in a simple way after each iteration, rather than to recompute it entirely as in the standard Newton method or leave it unchanged as in the modified Newton method. Here we consider the update, known as the Broyden–Fletcher–Goldfarb–Shanno (BFGS). It is most conveniently written in terms of K^{(t+1)} rather than K^{(t)} and has the form

[ K ^ {(i)} ] ^ {- 1} = [ I + w ^ {(i)} \{v ^ {(i)} \} ^ {T} ] [ K ^ {(i - 1)} ] ^ {- 1} [ I + v ^ {(i)} \{w ^ {(i)} \} ^ {T} ]. \tag {12.23}

The indicated matrix multiplications are never carried out in the computer implementation; instead v^{(i)} and w^{(i)} are stored and used only in computing the new search direction

\Delta \boldsymbol {d} ^ {(i)} = [ \boldsymbol {K} ^ {(i)} ] ^ {- 1} \psi (\boldsymbol {d} ^ {(i)}). \tag {12.24}

A line search of the form given in (12.21) is adopted. The BFGS formulae for v^{(i)} and w^{(i)} are

\boldsymbol {v} ^ {(t)} = \psi (\boldsymbol {d} ^ {(t)}) \left(1 + \alpha^ {(t - 1)} \left[ \frac {\{\Delta \boldsymbol {d} ^ {(t - 1)} \} ^ {T} \gamma^ {(t)}}{\{\delta^ {(t)} \} ^ {T} \{\psi (\boldsymbol {d} ^ {(t - 1)}) \}} \right] ^ {1 / 2}\right) - \psi (\boldsymbol {d} ^ {(t)}), \tag {12.25}

and

w ^ {(i)} = \frac {\delta^ {(i)}}{\{\delta^ {(i)} \} ^ {T} \gamma^ {(i)}}, \tag {12.26}

where

\delta^ {(i)} = d ^ {(i)} - d ^ {(i - 1)} = a ^ {(i - 1)} \Delta d ^ {(i - 1)},

and

\gamma^ {(i)} = \psi (d ^ {(i)}) - \psi (d ^ {(i - 1)}).

The method has been successfully implemented and used by Matthies and Strang ^{(91)} and Geradin and Hogge ^{(92)} for both static and transient dynamic nonlinear problems. The stability of BFGS with respect to unloading has been emphasised by Matthies and Strang. ^{(91)} A related method by Crisfield ^{(93)} also shows much promise.

Rather than work with the inverse of K^{(t)} as given in (12.23), Geradin and Hogge ^{(92)} work with the update formula

\boldsymbol {K} ^ {(t)} = \boldsymbol {K} ^ {(t - 1)} + \frac {\gamma^ {(i)} \left\{\gamma^ {(i)} \right\} ^ {T}}{\left\{\gamma^ {(i)} \right\} ^ {T} \delta^ {(i)}} - \frac {\left\{\boldsymbol {K} ^ {(t - 1)} \delta^ {(i)} \right\} \left\{\boldsymbol {K} ^ {(t - 1)} \delta^ {(i)} \right\} ^ {T}}{\left\{\delta^ {(i)} \right\} ^ {T} \boldsymbol {K} ^ {(t - 1)} \delta^ {(i)}}, \tag {12.27}

and use a frontal solution scheme.

12.5.3 Conjugate gradient methods

In the conjugate gradient ^{(94)} algorithm we take

\boldsymbol {d} ^ {(i + 1)} = \boldsymbol {d} ^ {(i)} + \alpha^ {(i)} \boldsymbol {\delta} ^ {(i)}, \tag {12.28}

where

\delta^ {(i)} = \psi (d ^ {(i)}) + \beta^ {(i)} \delta^ {(i - 1)}, \tag {12.29}

in which a^{(i)} is chosen using a line search with the criterion that the total potential energy \pi(d^{(i+1)}) should be minimised.

Initially, \beta^{(0)} is set to zero. We list two possible values for \beta^{(t)} :

(i) The Hestenes-Stiefel ^{(94)} (Fletcher-Reeves ^{(95)} ) algorithm

\beta^ {(i)} = \frac {\{\psi^ {(i)} \} ^ {T} \psi^ {(i)}}{\{\psi^ {(i - 1)} \} ^ {T} \psi^ {(i - 1)}}. \tag {12.30}

(ii) The Polak-Ribiere ^{(96)} algorithm

\beta^ {(i)} = \frac {\{\psi^ {(i)} \} ^ {T} \gamma^ {(i)}}{\{\psi^ {(i - 1)} \} ^ {T} \psi^ {(i - 1)}}. \tag {12.31}

The method, which requires modest computer core requirements, has been improved by scaling and other techniques. ^{(97-99)} The Conjugate–Newton method of Irons ^{(100)} is also a development of the basic conjugate gradient algorithm.

12.5.4 Other useful solution techniques

Among the remaining solution procedures, dynamic relaxation (DR) methods are quite popular. The main idea in DR originated from the observation that with about 90% of critical damping, an equivalent transient dynamic analysis rapidly converges to the steady state, static solution. Recent modifications ^{(101-103)} of the method have concentrated on finding improved replacements for the mass matrix M and the damping matrix C which are used in DR. Although DR methods are generally not as powerful as the various Newton and conjugate gradient methods, they require very little computer core storage and explicit transient dynamic programs such as DYNPAK, described in Chapter 10, can be rapidly modified to be used as DR solvers for ad hoc static problems when no other static program is available and results are urgently required.

It is usually difficult to decide on the form of load incrementation to adopt for elasto-plastic problems and exploratory analyses are often required. The work of Bergan and Soreide ^{(104)} in this area appears to be quite promising.

Schemes which work with local and global modes, several meshes or hierarchical representations ^{(105-111)} for the displacements may also prove to be of prime importance in nonlinear finite element equation solving.

12.6 Other enhancements in elasto-plastic analysis

12.6.1 Substructuring and boundary element methods

Economies can be made in the numerical solution of elasto-plastic problems by the use of substructuring techniques. A substructure analysis generally comprises the following steps. ^{(112)}

• Separate groups of elements within the solid are collectively identified as substructures as indicated in Fig. 12.10.
• For each substructure, the element stiffness matrices are assembled to give the global stiffness matrix of the substructure.
• The equations relating to the internal nodal points (i.e. nodes not on the boundary) are eliminated. This process is known as condensation.
• Solution of the system of resulting simultaneous equations is obtained by assembling all the individual substructures and any remaining elements which have not been associated with a substructure. This gives the nodal displacements and reactions for all nodal points on interfaces between substructures and for nodes of elements which are not related to any substructure.

- Return to the individual substructures to evaluate the displacements at interior nodes and finally obtain the element stresses.

text_image

substructure 1 II III

Fig. 12.10 Substructure analysis of elasto-plastic problems.

The very nature of the frontal equation solution process described in Section 6.4.12 makes the use of substructure techniques a simple affair, since, when the front has advanced into a structure to a certain position, the reduced frontal equations are essentially the condensed equations for a substructure corresponding to the part of the structure already considered.

For elasto-plastic problems, the part of the structure which (by physical considerations or experience!) is known to remain elastic during the deformation process can be defined as one substructure and the remaining elements considered individually. Thus during incremental/iterative solution the substructure stiffness will remain unaltered, for solution by the tangential stiffness method, and the substructure assembly and condensation process described above need be performed only once with an equation resolution process, necessitating only reduction of the R.H.S. terms being followed thereafter. The individual elements not associated with the substructure (and which model the elasto-plastic behaviour) are treated in the normal way as described in Chapter 7.

This approach can result in considerable computational economies, particularly if the mesh subdivision within the substructure is a fine one. It can be argued that a fine mesh subdivision is not warranted for regions where elastic behaviour is anticipated, but for structures which are to be subjected to more than one type of loading such an optimal mesh grading may not be possible. For example, with reference to Fig. 12.10, two separate loadings may cause plastic yielding in substructures II and III respectively and consequently a fine mesh grading within each of these regions cannot be avoided.

An extension of the above process is afforded by the use of the boundary integral method. ^{(113-115)} The boundary integral procedure requires trial functions which satisfy the governing equations directly and then attempt to satisfy the boundary conditions by a collocation, least-squares or Galerkin

procedure. In order to find trial functions which satisfy the governing equations we are, at present, generally confined to linear elastic situations. Thus for the solution of elasto-plastic problems a coupled approach can be employed ^{(113,115)} with the elastic region of the structure being modelled by boundary elements and conventional finite elements employed to treat the elasto-plastic zones. Such direct coupling leads to nonsymmetric matrices which is acceptable if the equation set is dominated by the boundary integral equations.

This approach promises efficient numerical solutions particularly for cases of limited yielding in three-dimensional solids where the surface area/volume ratio is relatively small. The process can also be used to advantage in infinite domain structures such as rock mass problems or soil/structure interaction problems with boundary elements being employed to model the exterior domain.

12.6.2 Interactive computing

The solution of elasto-plastic problems inevitably requires some degree of insight into the structural behaviour before choice of solution parameters, such as load increment sizes, can be made. Even then it is difficult, if not impossible, to specify the most suitable values of load increments, tolerance factors for each load case and also choice of the optimal solution process (e.g. initial stiffness, tangential stiffness or some combined algorithm) is equally difficult to arrive at.

To this end, the developments which are currently taking place in interactive computing will become increasingly important. Here we envisage the situation where the results for a particular load increment are held in core while the solution is scrutinized. Depending on the convergence characteristics, etc., the load increment size and convergence tolerance factor are then input and solution continued for a further increment. If required the nonlinear solution process can be redefined at this stage changing, for example, from a tangential stiffness to an initial stiffness algorithm if collapse conditions are being approached. Furthermore if the numerical process did not converge in the previous increment, the calculations could be repeated for a smaller load increment size or a different solution algorithm.

12.6.3 Computational techniques

Many new and improved programming strategies are developing in connection with finite element software and the interested reader is directed to the work of Schrem ^{(116,117)} and others ^{(118)} who are active in this area.

12.7 Concluding remarks

Throughout this text we have described numerical techniques and computer codes for a variety of engineering applications. Treatment has been limited to situations where the finite element method can be used to provide

nonlinear solutions with a measure of confidence. In this final chapter we have attempted to indicate some areas of further study and here the applicability to design problems is not so clear. For example, for soils and concrete some divergence of opinion still exists as to selection of an appropriate material model. Indeed at the present time it is true to say that numerical solution capabilities are in advance of the knowledge of fundamental material behaviour. This is particularly true for dynamic problems where there is a scarcity of information on material response under transient conditions. In this respect it would appear that nonlinear finite element methods offer the possibility of conducting ‘numerical experiments’ to provide insight on material behaviour which could not be obtained by experiment alone.

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