1. If a displacement field is described as follows, u = (−x+2y+6xy)10 and v = (3x + 6y Determine the strain components v xx , v yy , and v xy at the point x = 1; y = 0. 2. Explain briefly about the following: (a) Variational method. (b) Importance of Boundary cond itions. 3. Discuss the following basic principles of finite element method. (a) Derivation of element stiffness matrix. (b) Assembly of Global stiffness Matrix. 4.What are the various teps involved in finite Element method and explain them through an Example 5. Compare and contrast the “Rayleigh comment on both the methods. 6. What are the various approximate methods of anal ysis and exp 7. (a) Explain the advantages and disadvantages of Finite Element Method. (b) What is meant by total potential of elastic str uc ure? Write the expression for total potential of a cantilever beam with uniformly dist ributed 8. In a plane strain problem, we have σx = 137.90*10 6 Pa σy= -68.95*10 Determine the value of the stress

Finite Element Methods

Asymptotic analysis and Shishkin-type decomposition for an elliptic convection-diffusion problem

In this paper, variational problems with nonpolynomial growth are studied by means of Orlicz space theory. The necessarity and sufficiency condition of weakly sequential semicontinuity for variational integrals are obtained in the setting of Orlicz\|Sobolev space, which generalizes the corresponding results for variational problems with polynomial growth.

Semicontinuity Problems in the Calculus of Variations

In spite of the advances in computer technology, there is still a need for more computationally efficient methods for performing stress analysis. One approach which is receiving increasing attention is global/local analysis. Such analyses can take a variety of forms. The form described herein uses two distinct meshes (one global and one local), but retains the same level of accuracy as one would obtain if one was to use a single refined global mesh. The accuracy is retained by using an iterative procedure to enforce equilibrium between the global and local regions. The procedure was tested for two configurations. The good performance observed indicates that the iterative global/local procedure warrants further examination.

Iterative Global/Local Finite Element Analysis

We developa general multiscale method for coupling atomistic and continuum simulations using the framework of the heterogeneous multiscale method (HMM). Both the atomistic and the continuum models are formulated in the form of conservation laws of mass, momentum and energy. A macroscale solver, here the finite volume scheme, is used everywhere on a macrogrid; whenever necessary the macroscale fluxes are computed using the microscale model, which is in turn constrained by the local macrostate of the system, e.g. the deformation gradient tensor, the mean velocity and the local temperature. We discuss how these constraints can be imposed in the form of boundary conditions. When isolated defects are present, we developan additional strategy for defect tracking. This method naturally decoup les the atomistic time scales from the continuum time scale. Applications to shock propagation, thermal expansion, phase boundary and twin boundary dynamics are presented. r 2005 Elsevier Ltd. All rights reserved.

https://web.math.princeton.edu/~weinan/pdf%20files/mm%20dynamics.pdf

Multiscale modeling of the dynamics of solids at finite temperature

A periodic relaxation method for computing microstructures

An iso-parametric finite element method is introduced in this paper to study cavitations and configurational forces in nonlinear elasticity. The method is shown to be highly efficient in capturing the cavitation phenomenon, especially in dealing with multiple cavities of various sizes and shapes. Our numerical experiments verified and extended, for a class of nonlinear elasticity materials, the theory of Sivaloganathan and Spector on the configurational forces of cavities, as well as justified a crucial hypothesis of the theory on the cavities. Numerical experiments on configurational forces indicate that, in the case of a round reference configuration with radially symmetric stretch on the boundary, the cavitation centered at the origin is the unique energy minimizer. Numerical experiments also reveal an interesting size effect phenomenon: for macro-scale pre-existing-defects, the cavitation process is dominated by the relatively larger pre-existing-defects, and the cavitation tendency of much smaller pre-existing-defects is significantly suppressed.

/pdf/a-numerical-study-on-cavitation-in-nonlinear-elasticity-271vqfnwm5.pdf

A numerical study on cavitation in nonlinear elasticity — defects and configurational forces

A numerical method called element removal method is designed to calculate singular minimizers which cannot be approximated by simple applications of standard numerical methods because of the so-called Lavrentiev phenomenon. The convergence of the method is proved. The results of numerical experiments show that the method is effective.

Element removal method for singular minimizers in variational problems involving Lavrentiev phenomenon

A rotational transformation method and an incremental crystallization method are developed to overcome some of the difficulties involved in the computation of microstructures. The numerical method based on these techniques has proved to be convergent. To increase further the accuracy of the computation, a technique is applied to remove the boundary effect of the numerical solutions. Numerical results for a double well problem are given to show the efficiency of the techniques.

Rotational transformation method and some numerical techniques for computing microstructures

A numerical method, with truncation methods as a special case, for computing singular minimizers in variational problems is described. It is proved that the method can avoid Lavrentiev phenomenon and detect singular minimizers. The convergence of the method is also established. Numerical results on a 2-D problem are given.

/pdf/a-numerical-method-for-computing-singular-minimizers-2l2rm78enn.pdf

Zhiping Li

Papers

A periodic relaxation method for computing microstructures

A numerical study on cavitation in nonlinear elasticity — defects and configurational forces

Element removal method for singular minimizers in variational problems involving Lavrentiev phenomenon

Rotational transformation method and some numerical techniques for computing microstructures

A numerical method for computing singular minimizers