Lagrangian element

Using two-noded Lagrangian elements the shape functions are given as... 

Since the Lagrangian walls are impermeable, the mass of the Lagrangian element is constant. At time t, when the walls of the element are separated by an Eulerian distance dx, the density of the fluid within it must be... 

Two answers can be given by considering the thermodynamics of the cube. In one approach we consider a thermodynamic system whose boundaries coincide with those of the cube and move with the material in the other we consider a thermodynamic system whose boundaries coincide with the cube at some instant but remain fixed in space so that material flows across them (i.e., a Lagrangian element and an Eulerian element, respectively)—see Figure 11.10. 

For the Lagrangian element, the source of energy is the work done on the boundaries of the system as they move—with power a k -ek on the pair of faces normal to z and —a k ek on the pair normal to x (where energy is expended rather than absorbed) but what is the energy source for the thermodynamic system with stationary boundaries The boundaries being stationary, no work is done on this system the system is maintained in a state that is steady through time by fluxes of mass, inward across boundary planes normal to z and outward across bounding planes normal to x. Let... 

Rotation matrices may be viewed as an alternative to particles. This approach is based directly on the orientational Lagrangian (1). Viewing the elements of the rotation matrix as the coordinates of the body, we directly enforce the constraint Q Q = E. Introducing the canonical momenta P in the usual manner, there results a constrained Hamiltonian formulation which is again treatable by SHAKE/RATTLE [25, 27, 20]. For a single rigid body we arrive at equations for the orientation of the form[25, 27]... 

Lagrangian-Eulerian (ALE) method. In the ALE technique the finite element mesh used in the simulation is moved, in each time step, according to a predetermined pattern. In this procedure the element and node numbers and nodal connectivity remain constant but the shape and/or position of the elements change from one time step to the next. Therefore the solution mesh appears to move with a velocity which is different from the flow velocity. Components of the mesh velocity are time derivatives of nodal coordinate displacements expressed in a two-dimensional Cartesian system as... 

FINITE ELEMENT MODELLENG OF POLYMERIC FLOW PROCESSES 3.5.3 VOF method in Lagrangian frameworks... 

Donea, J., 1992. Arbitrary Lagrangian-Eulerian finite element methods. In Belytschko, T. and Hughes, T. J. R. (eds), Computational Methods for Transient Analysis, Elsevier Science, Amsterdam. 

Figure 5.4 The finite element mesh configurations in the Arbitrary Lagrangian-Eulerian scheme...

Figure 2.12. A flow tube used to derive one-dimensional flow equations in Lagrangian coordinates. Internal surfaces are massless, impermeable partitions to aid in visualizing elements of fluid in Lagrangian coordinates.

Arbitrary-Lagrangian-Eulerian (ALE) codes dynamically position the mesh to optimize some feature of the solution. An ALE code has tremendous flexibility. It can treat part of the mesh in a Lagrangian fashion (mesh velocity equation to particle velocity), part of the mesh in an Eulerian fashion (mesh velocity equal to zero), and part in an intermediate fashion (arbitrary mesh velocity). All these techniques can be applied to different parts of the mesh at the same time as shown in Fig. 9.18. In particular, an element can be Lagrangian until the element distortion exceeds some criteria when the nodes are repositioned to minimize the distortion. 

Another alternative for such problems that has become widely available in Lagrangian finite element codes is the eroding slide line option (see, e.g., [66]). 

With this approach, when an element becomes severly distorted, it is eliminated from the computational grid and becomes a free mass point. Clearly, care must be taken to avoid eliminating elements that could potentially influence the problem at some later time. An example of a three-dimensional Lagrangian calculation that uses the eroding element scheme is presented in the next section. 

With the advances in computing hardware that have occurred over the last decade, three-dimensional computational analyses of shock and impact problems have become relatively common. In Lagrangian calculations, element erosion schemes have provided a means for handling the large deformations and material failure that is often involved, and Fig. 9.28 shows results of a penetration calculation which makes use of this methodology [68]. 

J. Donea, Arbitrary Lagrangian-Eulerian Finite Element Methods, Computational Methods for Transient Analysis (edited by J.D. Achenbach), North-Holland, Amsterdam, 1983. 

Goudreau, G.L. and Hallquist, J.O., Synthesis of Hydrocode and Finite Element Technology for Large Deformation Lagrangian Computation, Lawrence Livermore Laboratory, University of California Preprint No. UCRL-82858, Livermore, CA, 13 pp., August 1979. 

The term macromixing refers to the overall mixing performance in a reactor. It is usually described by the residence time distribution (RTD). Originally introduced by Danckwerts (1958), this concept is based on a macroscopic lumped population balance. A fluid element is followed from the time at which it enters the reactor (Lagrangian viewpoint - observer moves with the fluid). The probability that the fluid element will leave the reactor after a residence time t is expressed as the RTD function. This function characterises the scale of mixedness in a reactor. 

In simulating physical operations carried out in stirred vessels, generally one has the choice between a Lagrangian approach and a Eulerian description. While the former approach is based on tracking the paths of many individual fluid elements or dispersed-phase particles, the latter exploits the continuum concept. The two approaches offer different vistas on the operations and require different computational capabilities. Which of the two approaches is most... 

For isothermal, first-order chemical reactions, the mole balances form a system of linear equations. A non-ideal reactor can then be modeled as a collection of Lagrangian fluid elements moving independe n tly through the system. When parameterized by the amount of time it has spent in the system (i.e., its residence time), each fluid element behaves as abatch reactor. The species concentrations for such a system can be completely characterized by the inlet concentrations, the chemical rate constants, and the residence time distribution (RTD) of the reactor. The latter can be found from simple tracer experiments carried out under identical flow conditions. A brief overview of RTD theory is given below. 

The PFR model ignores mixing between fluid elements at different axial locations. It can thus be rewritten in a Lagrangian framework by substituting a = Tpfrz, where a denotes the elapsed time (or age) that the fluid element has spent in the reactor. At the end of the PFR, all fluid elements have the same age, i.e., a = rpfr. Moreover, at every point in the PFR, the species concentrations are uniquely determined by the age of the fluid particles at that point through the solution to (1.2). 

In the CRE literature, the residence time distribution (RTD) has been shown to be a powerful tool for handling isothermal first-order reactions in arbitrary reactor geometries. (See Nauman and Buffham (1983) for a detailed introduction to RTD theory.) The basic ideas behind RTD theory can be most easily understood in a Lagrangian framework. The residence time of a fluid element is defined to be its age a as it leaves the reactor. Thus, in a PFR, the RTD function E(a) has the simple form of a delta function ... 

A Lagrangian description of the velocity field can be used to find the location X(f) of the fluid element at time 0 < t that started at X(0). In the Lagrangian description, (3.3) implies that the scalar field associated with the fluid element will remain unchanged, i.e., (X(f), 0 = (X(0), 0). 

In a Lagrangian PDF simulation, each notional particle represents a fluid element with mass u)wAm. In a constant-density system, the unit mass is defined by... 

We have seen that the quark mass dependence of ferromagnetism should be important, while we have treated it as an input parameter. When we consider the realization of chiral symmetry in QCD, the quark mass should be dynamically generated as a result of the vacuum superconductivity qq pairs are condensed in the vacuum. We consider here SU(2)l x SU(2)r symmetry. Then Lagrangian should be globally invariant under the operation of any group element with constant parameters, except the symmetry-breaking term... 

The deviation of the diagonal Lagrangian-multipliers (see Eq. 6, 7) obtained for the orbitals after the given transformation from the canonical diagonal Fock-matrix elements. 

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