Incompatible Displacements Research Articles

Energy preservation and entropy in Lagrangian space- and time-staggered hydrodynamic schemes

Usual space- and time-staggered (STS) leap-frog Lagrangian hydrodynamic schemes-such as von Neumann-Richtmyer's (1950), Wilkins' (1964), and their variants-are widely used for their simplicity and robustness despite their known lack of exact energy conservation. Since the seminal work of Trulio and Trigger (1950) and despite the later corrections of Burton (1991), it is generally accepted that these schemes cannot be modified to exactly conserve energy while retaining all of the following properties: STS stencil with velocities half-time centered with respect to positions, explicit second-order algorithm (locally implicit for internal energy), and definite positive kinetic energy.It is shown here that it is actually possible to modify the usual STS hydrodynamic schemes in order to be exactly energy-preserving, regardless of the evenness of their time centering assumptions and retaining their simple algorithmic structure. Burton's conservative scheme (1991) is found as a special case of time centering which cancels the term here designated as incompatible displacements residue. In contrast, von Neumann-Richtmyer's original centering can be preserved provided this residue is properly corrected. These two schemes are the only special cases able to capture isentropic flow with a third order entropy error, instead of second order in general.The momentum equation is presently obtained by application of a variational principle to an action integral discretized in both space and time. The internal energy equation follows from the discrete conservation of total energy. Entropy production by artificial dissipation is obtained to second order by a prediction-correction step on the momentum equation.The overall structure of the equations (explicit for momentum, locally implicit for internal energy) remains identical to that of usual STS leap-frog schemes, though complementary terms are required to correct the effects of time-step changes and artificial viscosity updates.In deriving these schemes, an apparently novel approach of flux-in-time was introduced to correct numerical residues and ensure energy conservation. This method can be applied to essentially any numerical scheme whenever required or desired provided space and time numerical consistency is preserved.Numerical test cases are presented confirming the conservative character of the new CSTS schemes down to computer round-off errors, and showing various improvements compared to the standard von Neumann-Richtmyer and Wilkins STS schemes, mostly on shock levels, shock velocities, singularity induced distortions, and CFL stability limits.

Automated identification and indexing of dislocations in crystal interfaces

We present a computational method for identifying partial and interfacial dislocations in atomistic models of crystals with defects. Our automated algorithm is based on a discrete Burgers circuit integral over the elastic displacement field and is not limited to specific lattices or dislocation types. Dislocations in grain boundaries and other interfaces are identified by mapping atomic bonds from the dislocated interface to an ideal template configuration of the coherent interface to reveal incompatible displacements induced by dislocations and to determine their Burgers vectors. In addition, the algorithm generates a continuous line representation of each dislocation segment in the crystal and also identifies dislocation junctions.

Beam and shell element model for advanced analysis of steel structural members

Plastic zone method of advanced analysis, which uses shell elements to model the entire structure, is the most accurate method available to predict the ultimate strength and behavior of steel frames. The disadvantage of such full shell plastic zone models is that it is computationally expensive and hence its use is limited to small structures. Beam elements in commercial finite element packages can model residual stress and capture spread of plasticity, but cannot model local buckling of plates that the member is made up of, which leads to unloading and failure in steel frames. A hybrid model using shell elements only in the regions vulnerable to elastic or inelastic local buckling and beam elements in other locations could overcome this limitation of full beam element model. The issues in using this hybrid model are, knowing a priori the location and length of the shell element region and connecting the beam and shell regions without any artificial stress concentrations or incompatible displacements. In this study, in addition to addressing these issues, the hybrid model is systematically evaluated by studying its performance in structural elements. It is seen that the hybrid model strength predictions has an average error of only 0.91% but requires on an average 83% less computational time when compared to the full shell plastic zone models.

A component mode synthesis algorithm for multibody dynamics of wind turbines

A system reduction scheme related to a multibody formulation of wind turbine dynamics is devised. Each substructure is described in its own frame of reference, which is moving freely in the vicinity of the moving substructure, in principle without any constraints to the rigid body part of the motion of the substructure. The system reduction is based on a component mode synthesis method, where the response of the internal degrees of freedom of the substructure is described as the quasi-static response induced by the boundary degrees of freedom via the constraint modes superimposed in combination to a dynamic component induced by inertial effects and internal loads. The latter component is modelled by a truncated modal expansion in fixed interface undamped eigenmodes. The selected modal vector base for the internal dynamics ensures that the boundary degrees of freedom account for the rigid-body dynamics of the substructure, and explicitly represent the coupling degrees of freedom at the interface to the adjacent substructures. The method has been demonstrated for a blade structure, which has been modelled as two substructures. Two modelling methods have been examined where the first is by use of fixed–fixed eigenmodes for the innermost substructure and fixed–free eigenmodes for the outermost substructure. The other approach is by use of fixed–free eigenmodes for both substructures. The fixed–fixed method shows good correspondence with the full FE model which is not the case for the fixed–free method due to incompatible displacements and rotations at the interface between the two substructures. Moreover, the results from the reduced model by use of constant constraint modes and constant fixed interface modes over a large operating area for the wind turbine blade are almost identical to the full FE model.

Two mixed finite element formulations with area bubble functions for tetrahedral elements

We address the solution of geometrically linear elastic problems for three-dimensional solid mechanics using tetrahedral finite elements. Starting from a five field weak formulation involving fields for compatible displacements, incompatible displacements, pressure, enhanced strains and stresses, both, the mixed method of incompatible modes and the mixed method of enhanced strains are considered as special cases. As a key idea, area bubble functions are used for both mixed finite element formulations in order to enrich the displacement field and the enhanced strain field, respectively. Appropriate conditions for satisfaction of the patch test and unique solution of the discrete equation are verified. We also observe that, contrary to a the standard enhanced assumed strain method applied to triangles and tetrahedra our mixed formulations with pressure as an independent variable are not degenerate. In the representative examples firstly a numerical verification of the patch test is obtained. Two additional examples, Cook’s membrane problem and a plate with a ring hole, illustrate the good performance of the presented approaches in comparison to existing finite element formulations.

The presence, characteristics and earthquake implications of the St. Lawrence fault zone within and near Lake Ontario (Canada–USA)

Questions persist concerning the earthquake potential of the populous and industrial Lake Ontario (Canada–USA) area. Pertinent to those questions is whether the major fault zone that extends along the St. Lawrence River valley, herein named the St. Lawrence fault zone, continues upstream along the St. Lawrence River valley at least as far as Lake Ontario or terminates near Cornwall (Ontario, Canada)–Massena (NY, USA). New geological studies uncovered paleotectonic bedrock faults that are parallel to, and lie within, the projection of that northeast-oriented fault zone between Cornwall and northeastern Lake Ontario, suggesting that the fault zone continues into Lake Ontario. The aforementioned bedrock faults range from meters to tens of kilometers in length and display kinematically incompatible displacements, implying that the fault zone was periodically reactivated in the study area. Beneath Lake Ontario the Hamilton–Presqu'ile fault lines up with the St. Lawrence fault zone and projects to the southwest where it coincides with the Dundas Valley (Ontario, Canada). The Dundas Valley extends landward from beneath the western end of the lake and is marked by a vertical stratigraphic displacement across its width. The alignment of the Hamilton–Presqu'ile fault with the St. Lawrence fault zone strongly suggests that the latter crosses the entire length of Lake Ontario and continues along the Dundas Valley. The Rochester Basin, an east–northeast-trending linear trough in the southeastern corner of Lake Ontario, lies along the southern part of the St. Lawrence fault zone. Submarine dives in May 1997 revealed inclined layers of glaciolacustrine clay along two different scarps within the basin. The inclined layers strike parallel to the long dimension of the basin, and dip about 20° to the north–northwest suggesting that they are the result of rigid-body rotation consequent upon post-glacial faulting. Those post-glacial faults are growth faults as demonstrated by the consistently greater thickness, unit-by-unit, of unconsolidated sediments on the downthrown (northwest) side of the faults relative to their counterparts on the upthrown (southeast) side. Underneath the western part of Lake Ontario is a monoclinal warp that displaces the glacial and post-glacial sediments, and the underlying bedrock–sediment interface. Because of the post-glacial growth faults and the monoclinal warp the St. Lawrence fault zone is inferred to be tectonically active beneath Lake Ontario. Furthermore, within the lake it crosses at least five major faults and fault zones and coexists with other neotectonic structures. Those attributes, combined with the large earthquakes associated with the St. Lawrence fault zone well to the northeast of Lake Ontario, suggest that the seismic risk in the area surrounding and including Lake Ontario is likely much greater than previously believed.

An improved hybrid-stress element approach to torsion of shafts

A four-noded torsional element with only four stress parameters is developed based on the Hellinger–Reissner principle by optimizing element trial stresses through the introduction of incompatible displacements. Two boundary hybrid-stress elements with six stress parameters are also developed to meet the traction-free condition on one or two adjacent sides of a four-noded element, respectively. Numerical results show that these elements give accurate results for the warping displacement, the angle of twist per unit length, as well as the shear stresses for shafts with a polygonal cross-section. Comparison with alternative displacement and stress elements shows that the present elements give results of the same accuracy with a coarser mesh. The elements are shown to be insensitive to mesh distortion and to exhibit rapid convergence with h-refinement.

New incompatible four-noded axisymmetric elements with assumed strains

This paper introduces two four-node quadrilateral axisymmetric elements with assumed strains. Both of them are formulated by the modified Hellinger-Reissner principle to employ incompatible displacements and assumed strains. In the formulation of the first one, the strains resulting from the incompatible displacement are corrected to pass the constant strain patch test. The second one contains the incompatible functions that are obtained directly from the element boundary condition of internal displacements. The elements are evaluated on the several problems of bending and material incompressibility with regular and distorted meshes. The results show that the new elements perform excellently in deformation and stress calculation.

New axisymmetric solid elements based on an extended Hellinger-Reissner principle

4-node incompatible axisymmetric hybrid stress elements are developed by a modified rational approach based on an extended Hellinger-Reissner principle. The stress fields are derived by using the constraint conditions that the equilibrium equations of higher order stress terms are imposed in a variational sense through the additional incompatible displacements. The resulting elements enjoy good performance in bending problems and at the nearly incompressible limit. Several examples are given to demonstrate the performance of the new elements.

A unified theory for formulation of hybrid stress membrane elements

AbstractA theory is proposed in this paper to explain, in a unified manner, the three rational approaches using the incompatible displacements uλ in the hybrid element formulation. Limited to the plane stress case, the theory suggests that the introduction of uλ is to constrain the total value of the invariant I2(= σxσy – τ) of the assumed stresses over the element to zero. In the plane stress case, this is the well‐known condition for the stress distributions to be independent of the elastic constants of the material. The theory proposed is also used to explain the closeness of the performances of the four 4‐node hybrid stress membrane elements, 5β‐I, 5β‐II, 5β‐A and 5β‐C, that have been developed previously. The differences among them are shown to be of the second degree in the distortion measures of the element shape, when they are compared on the basis of the non‐vanishing total value of I2 of their respectively adopted assumed stress field. Several numerical examples are used to verify and to illustrate the proposed theory.

A quadrilateral hybrid stress element for Mindlin plates based on incompatible displacements

AbstractAn hybrid stress element formulation based on internal, incompatible displacements is used to develop efficient Mindlin plate elements. The 4‐node quadrilateral Mindlin plate element is derived from a modified energy functional. Both displacements and stresses are defined in the natural co‐ordinate interpolation system. The assumed stress field is obtained by tensor transformation and so chosen as to ensure that the element is co‐ordinate invariant and stable. Shear locking is avoided through an appropriate identification of the internal, incompatible displacement field. The role played by incompatible displacements in the formulation of hybrid stress elements for thin and moderately thick plates is discussed. Numerical applications are presented to illustrate the accuracy and reliability of the suggested Mindlin plate element.

New strategy for assumed stresses for 4‐node hybrid stress membrane element

AbstractAn oblique (skew) co‐ordinate system defined at the origin of the isoparametric co‐ordinates is introduced as the reference co‐ordinate system for assumed stresses in the formulation of a 4‐node hybrid membrane element. The initially uncoupled stresses in these reference co‐ordinates can be constrained to satisfy the equilibrium conditions pointwise by using the procedure of Pian and Wu7, when suitable incompatible displacements are introduced. Such strategy allows the element stiffness to be formulated by using either the Hellinger‐Reissner or the complementary energy functional. Numerical results show that the performances of three elements, constructed by using different sets of incompatible displacements in this paper, are practically identical and are all of slightly higher accuracy than that of the Pian‐Sumihara element6.

The hybrid stress model for mindlin-reissner plates based on a stress optimization condition

A stress optimization condition for hybrid stress finite elements is applied to develop a Mindlin-Reissner plate element. A modified Hellinger-Reissner energy functional, derived from a stress optimization condition, is employed to relax the patch test constraint of incompatible displacements. An eight-node serendipity thin and moderately thick plate element, which is stable and free from shear locking, is formulated from the optimized stress distribution and the use of incompatible displacements. Numerical results indicate that the resulting element is accurate in both displacements and stresses, and insensitive to mesh distortion.

Crack propagation in brittle materials under compressive stresses studied by caustics

An experimental study was undertaken concerning the propagation of a slant crack under compression. By using the equations of the deformed shape of a crack and by introducing a correction model in order to prevent the crack lips from incompatible displacements, an estimate of the stress distribution along the crack borders was achieved. It was found that an opening-mode K1 stress intensity factor (SIF) must be introduced at the crack tips in the overall compressive stress field, in order to give the required space for the lip-slip phenomenon, due to shearing, to take place. This local dilatation in the vicinity of the crack tip, together with the lip-slip phenomenon, due to which the initial crack tip is displaced into a new position along the deformed crack borders, causes the out-of-plane propagation of the crack, either towards the largest compressive stress in the case of biaxial stress field, or towards the applied compression in a uniaxial compressive field. A series of experiments on PMMA (perspex) rectangular specimens with pre-existed cracks and slits has been executed and the type of the stress field in the front of the propagated branches is examined by using the method of caustics. It was found that the crack propagation under compression is an interactive process of two conjugate branches, which is strongly influenced by the boundary conditions of the pre-existed discontinuity.

A rational formulation of isoparametric hybrid stress elements for three-dimensional stress analysis

Based on the recently proposed method of energy orthogonality enforcement between the stresses and derived strains from the incompatible displacements, three 8-noded hexahedral hybrid stress elements are derived. Stress interpolants for the elements are expressed in terms of uncoupled functions of the natural coordinates, thus Pian's efficient scheme for inverting the complementary energy matrix is applicable. The choice of energy orthoganality instead of equilibrium as a means to eliminate the parasitic stress components not only resolved the commonly encountered problem of rank deficiency but also presents a more rational approach. Numerical examples are given and the achieved accuracy is satisfactory. All the elements are capable of handling nearly incompressible materials. An optical choice is made by further examining the elements' perfommances in a thin plate problem.

A rational approach for choosing stress terms for hybrid finite element formulations

AbstractA new approach for choosing the stress terms for a hybrid stress element is based on the condition of vanishing of the virtual work along the element boundary due to the stress terms higher than constant and the additional incompatible displacement. Examples using 4‐node plane stress elements have shown that when the incompatible displacements also satisfy the constant strain patch test the resulting elements will provide the most accurate solutions. Advantages of this approach for the formulation of an axisymmetric solid are also indicated.