David J. Benson

Computational methods in Lagrangian and Eulerian hydrocodes

Explicit finite element and finite difference methods are used to solve a wide variety of transient problems in industry and academia. Unfortunately, explicit methods are rarely discussed in detail in finite element text books. This paper reviews the basic explicit finite element and finite difference methods that are currently used to solve transient, large deformation problems in solid mechanics. A special emphasis has been placed on documenting methods that have not been previously published in journals.

An efficient, accurate, simple ALE method for nonlinear finite element programs

Abstract A simple arbitrary Lagrangian-Eulerian formulation is presented. An operator split separates the Lagrangian and Eulerian processes, allowing a Lagrangian finite element program to be extended to this formulation with little difficulty. Example problems illustrate the strengths and weaknesses of the formulation.

ANALYTICAL AND COMPUTATIONAL DESCRIPTION OF EFFECT OF GRAIN SIZE ON YIELD STRESS OF METALS

Four principal factors contribute to grain-boundary strengthening: (a) the grain boundaries act as barriers to plastic flow; (b) the grain boundaries act as dislocation sources; (c) elastic anisotropy causes additional stresses in grain-boundary surroundings; (d) multislip is activated in the grain-boundary regions, whereas grain interiors are initially dominated by single slip, if properly oriented. As a result, the regions adjoining grain boundaries harden at a rate much higher than grain interiors. A phenomenological constitutive equation predicting the effect of grain size on the yield stress of metals is discussed and extended to the nanocrystalline regime. At large grain sizes, it has the Hall-Petch form, and in the nanocrystalline domain the slope gradually decreases until it asymptotically approaches the flow stress of the grain boundaries. The material is envisaged as a composite, comprised of the grain interior, with flow stress sfG, and grain boundary work-hardened layer, with flow stress sfGB. The predictions of this model are compared with experimental measurements over the mono, micro, and nanocrystalline domains. Computational predictions are made of plastic flow as a function of grain size incorporating differences of dislocation accumulation rate in grain- boundary regions and grain interiors. The material is modeled as a monocrystalline core surrounded by a mantle (grain-boundary region) with a high work hardening rate response. This is the first computational plasticity calculation that accounts for grain size effects in a physically-based manner. A discussion of statisti- cally stored and geometrically necessary dislocations in the framework of strain-gradient plasticity is intro- duced to describe these effects. Grain-boundary sliding in the nanocrystalline regime is predicted from calcu- lations using the Raj-Ashby model and incorporated into the computations; it is shown to predispose the material to shear localization.  2001 Published by Elsevier Science Ltd on behalf of Acta Materialia Inc.

A single surface contact algorithm for the post-buckling analysis of shell structures

Abstract In some of our applications we are interested in how a structure behaves after it buckles. When a structure collapses completely, a single surface may buckle enough that it comes into contact with itself. The traditional approach of defining master and slave contact surfaces will not work because we do not know a priori how to partition the surface of the structure. This paper presents a contact algorithm that requires only a single surface definition for its input.

Momentum advection on a staggered mesh

Abstract Eulerian and ALE (arbitrary Lagrangian-Eulerian) hydrodynamics programs usually split a timestep into two parts. The first part is a Lagrangian step, which calculates the incremental motion of the material. The second part is referred to as the Eulerian step, the advection step, or the remap step, and it accounts for the transport of material between cells. In most finite difference and finite element formulations, all the solution variables except the velocities are cell-centered while the velocities are edge- or vertex-centered. As a result, the advection algorithm for the momentum is, by necessity, different than the algorithm used for the other variables. This paper reviews three momentum advection methods and proposes a new one. One method, pioneered in YAQUI, creates a new staggered mesh, while the other two, used in SALE and SHALE, are cell-centered. The new method is cell-centered and its relationship to the other methods is discussed. Both pure advection and strong shock calculations are presented to substantiate the mathematical analysis. From the standpoint of numerical accuracy, both the staggered mesh and the cell-centered algorithms can give good results, while the computational costs are highly dependent on the overall architecture of a code.

Volume of fluid interface reconstruction methods for multi-material problems

Butadiene polymers containing 70% or more of 1,2-structure and a relatively low melting point are produced by polymerizing 1,3-butadiene in the presence of a catalyst which has been prepared by admixing (A) an organic solvent solution containing 1,3-butadiene, a cobalt compound and an organoaluminium compound; (B) an amide compound of the formula (2) or (3): (2) or (3) wherein R1, R2 and R3 are respectively an H atom, aliphatic hydrocarbon radical of 1 to 7 carbon atoms or aromatic hydrocarbon radical of 6 or 7 carbon atoms, R3 is H or an aliphatic hydrocarbon radical of 1 to 3 carbon atoms and n is 2 to 5, and; (C) carbon disulfide.

A mixture theory for contact in multi-material Eulerian formulations

Abstract Multi-material Eulerian methods are attractive for solving a broad range of highly nonlinear problems in solid mechanics because they permit arbitrarily large deformations and automatically allow new free surfaces to evolve. Many of these problems involve the contact and separation of surfaces. There is an extensive literature for including contact inequality constraints in Lagrangian finite element methods, but little consideration has been given to contact in Eulerian methods. The contact forces between adjacent bodies are governed by the mixture theory in the Eulerian formulation, making the mixture theory the natural starting point in the treatment of the contact inequalities. A mixture theory that equilibrates the traction across the material interface and limits the shear stress according to a Coulomb friction law is developed.

Free-Surface Flow and Fluid-Object Interaction Modeling With Emphasis on Ship Hydrodynamics

Abstract : This paper presents our approach for the computation of free-surface/rigid-body interaction phenomena with emphasis on ship hydrodynamics. We adopt the level set approach to capture the free-surface. The rigid body is described using six-degree-of-freedom equations of motion. An interface-tracking method is used to handle the interface between the moving rigid body and the fluid domain. An Arbitrary Lagrangian Eulerian version of the residual-based variational multiscale formulation for the Navier Stokes and level set equations is employed in order to accommodate the fluid domain motion. The free-surface/rigid body problem is formulated and solved in a fully coupled fashion. The numerical results illustrate the accuracy and robustness of the proposed approach.

A multi-material Eulerian formulation for the efficient solution of impact and penetration problems

An Eulerian formulation is necessary for the accurate solution of contact and impact problems involving penetration and fracture. The Eulerian mesh is fixed in space, thereby eliminating all the problems associated with a distorted mesh that are commonly encountered with a Lagrangian formulation. Since the material flows through the mesh, additional data is necessary in an Eulerian formulation to describe the current contents of an element and additional calculations must be performed to update the data. The additional calculations, which account for the material transport between the elements, are usually much more expensive than the Lagrangian terms in the calculation. As a consequence, Eulerian calculations have been restricted to hypervelocity impacts, which cannot be solved in any other manner. This paper discusses strategies for restructuring the transport calculations so that the Eulerian formulation may be applied to a broader range of problems in science and engineering. Example calculations, performed on a workstation, are presented to demonstrate the efficiency of the proposed strategies.

On the effect of grain size on yield stress: extension into nanocrystalline domain

Abstract Four principal factors contribute to grain-boundary strengthening: (a) the grain boundaries act as barriers to plastic flow; (b) the grain boundaries act as dislocation sources; (c) elastic anisotropy causes additional stresses in grain-boundary surroundings; (d) multislip is activated in the grain-boundary regions, whereas grain interiors are initially dominated by single slip, if properly oriented. As a result, the regions adjoining grain boundaries harden at a rate much higher than grain interiors. A phenomenological constitutive equation predicting the effect of grain size on the yield stress of metals is discussed and extended to the nanocrystalline regime. At large grain sizes, it has the Hall–Petch form, and in the nanocrystalline domain the slope gradually decreases until it asymptotically approaches the flow stress of the grain boundaries. The material is envisaged as a composite, comprised of the grain interior, with flow stress σfB and grain boundary work-hardened layer, with flow stress σfGB. The predictions of this model are compared with experimental measurements over the mono, micro, and nanocrystalline domains. Computational predictions are made of plastic flow as a function of grain size incorporating differences of dislocation accumulation rate in grain-boundary regions and grain interiors. The material is modeled as a monocrystalline core surrounded by a mantle (grain-boundary region) with a high work hardening rate response. This is the first computational plasticity calculation that accounts for grain size effects in a physically-based manner.