J. Bonet

Variational and momentum preservation aspects of Smooth Particle Hydrodynamic formulations

This paper presents a new variational framework for various existing Smooth Particle Hydrodynamic (SPH) techniques and presents a new corrected SPH formulation. The linear and angular momentum preserving properties of SPH formulations are also discussed. The paper will show that in general in order to preserve angular momentum, the SPH equations must correctly evaluate the gradient of a linear velocity field. A corrected algorithm that combines kernel correction with gradient correction is presented. The paper will illustrate the theory presented with several examples relating to simple free surface flows.

Correction and stabilization of smooth particle hydrodynamics methods with applications in metal forming simulations

Smooth particle hydrodynamics (SPH) is a robust and conceptually simple method which suffers from unsatisfactory performance due to lack of consistency. The kernel function can be corrected to enforce the consistency conditions and improve the accuracy. For simplicity in this paper the SPH method with the corrected kernel is referred to as corrected smooth particle hydrodynamics (CSPH). The numerical solutions of CSPH can be further improved by introducing an integration correction which also enables the method to pass patch tests. It is also shown that the nodal integration of this corrected SPH method suffers from spurious singular modes. This spatial instability results from under integration of the weak form, and it is treated by a least-squares stabilization procedure which is discussed in detail in Section 4. The effects of the stabilization and improvement in the accuracy are illustrated via examples. Further, the application of CSPH method to metal-forming simulations is discussed by formulating the governing equation associated with the process. Finally, the numerical examples showing the effectiveness of the method in simulating metal-forming problems are presented. Copyright

A simple orthotropic, transversely isotropic hyperelastic constitutive equation for large strain computations

Abstract This paper presents a simple transversely isotropic hyperelastic constitutive equation that can be used to model fiber oriented elastic materials in the fully nonlinear range. The hyperelastic strain energy function that defines this material is given in terms of three new material parameters and equations relating these material parameters to the Poisson ratio and the Young modules of the material along the fiber direction and on the orthogonal plane are derived. Expressions for the second Piola—Kirchhoff tensor, the Cauchy stress tensor and the Lagrangian and Eulerian elasticity tensors are also obtained. Static and dynamic applications of this material are used to illustrate its performance.

Finite element analysis of the superplastic forming of thick sheet using the incremental flow formulation

The paper discusses the finite element analysis of the superplastic forming of thick sheet components. The incremental formulation proposed is based on a geometrical approximation of the flow type of constitutive equations that describe the behaviour of the alloy during forming. The spatial discretization is achieved using eight-noded finite elements. An algorithm capable of predicting the correct forming pressure is also presented in a form consistent with the incremental flow formulation. Some experimental validation of these techniques will be shown together with a number of more realistic applications which will illustrate the generality of these techniques and their ability to simulate the forming of complex components. Most of the material in this section is standard but has been included for the purpose of completeness and to introduce the reader to the notation used in the paper.

Alternative techniques for casting process simulation

Over the past 20 years, casting process simulation has been an active area of research. The simulation techniques are either based on solving governing partial differential equations using numerical schemes such as the finite element or finite difference methods, or a variety of heuristically based geometry driven methods. Numerical methods are more accurate, but geometry driven methods are computationally less expensive. This paper explores two alternative techniques to overcome some of the limitations of traditional numerical simulation schemes for the casting process simulation. The first technique uses a geometric transformation method known as the medial axis transformation, to predict hot spots whereas the second technique, based on meshless methods, is used for simulating the mould filling process.

The Efficient Computation of Bounds for Functionals of Finite Element Solutions in Large Strain Elasticity

We present an implicit a posteriori finite element procedure to compute bounds for functional outputs of finite element solutions in large strain elasticity. The method proposed relies on the existence of a potential energy functional whose local minima, over a space of suitably chosen continuous functions, corresponds to the problem solution. The output of interest is cast as a constrained minimization problem over an enlarged discontinuous finite element space. A Lagrangian is formed were the multipliers are an adjoint solution, which enforces equilibrium, and hybrid fluxes, which constrain the solution to be continuous. By computing approximate values for the multipliers on a coarse mesh, strict upper and lower bounds for the output of interest on a suitably refined mesh, are obtained. This requires a minimization over a discontinuous space, which can be carried out locally at low cost. The computed bounds are uniformly valid regardless of the size of the underlying coarse discretization. The method is demonstrated with two applications involving large strain plane stress incompressible neo-Hookean hyperelasticity.

Computer Simulation of Superplastic Forming in Restorative Dentistry

Superplastic forming of dental prostheses using Ti-6Al-4V alloy is a recent innovation. Such prostheses are lightweight and strong and can be produced using low cost dies made from standard dental casting investment materials. In contrast to the volume production of SPF components a dental prostheses is unique to each patient and the SPF pressure cycle required to form the prosthesis equally unique. Consequently it is necessary to employ computer simulation to reduce trial and error experimentation. This paper will summarise the computer simulation of the process and subsequent experimental validation. Both geometric and finite element simulation methods will be discussed with an emphasis on the pressure cycle algorithm contained in the latter method.

Finite element modelling of thin metal sheet forming

Abstract: This chapter reviews the finite element simulation of superplastic forming processes. Both the traditional flow formulation and the incremental flow formulation are presented. The chapter reviews the finite element discretisation of the equilibrium equations describing the motion of a forming sheet including available strategies for the evaluation of a correct forming pressure, which is often one of the key outputs of the simulation. A number of examples are given in relation to applications in the aerospace industry as well as more recent applications in the biomedical field.