scholarly journals A Mixed Finite Element-Based Numerical Method for Elastodynamics Considering Adhesive Interface Damage for Dynamic Fracture

2021 ◽  
pp. 2150010
Author(s):  
Christos G. Panagiotopoulos ◽  
Chrysoula Tsogka
Keyword(s):  
Acoustic Emission ◽  
Boundary Conditions ◽  
Seismic Waves ◽  
Mixed Finite Element ◽  
Mixed Finite Elements ◽  
Dynamic Rupture ◽  
Initiation And Propagation ◽  
Space Discretization ◽  
Type Boundary ◽  
The Stability

The numerical solution of the elastodynamic problem with kinematic boundary conditions is considered using mixed finite elements for the space discretization and a staggered leap-frog scheme for the discretization in time. The stability of the numerical scheme is shown under the usual CFL condition. Using the general form of Robin-type boundary conditions some cases of debonding and the resulting acoustic emission are studied. The methodology presented finds applications to geophysics such as in seismic waves simulation with dynamic rupture and energy release. In this paper, we focus on problems of fracture occurring at the interface of composite materials. Our goal is to study both the mechanism of crack initiation and propagation, as well as the acoustic emission of the released structure-borne energy which may be used in structural health monitoring and prognosis applications.

scholarly journals Interior penalty mixed finite element methods of any order in any dimension for linear elasticity with strongly symmetric stress tensor

2017 ◽  
Vol 27 (14) ◽  
pp. 2711-2743 ◽  
Author(s):  
Shuonan Wu ◽  
Shihua Gong ◽  
Jinchao Xu
Keyword(s):  
Error Estimate ◽  
Linear Elasticity ◽  
Mixed Finite Element ◽  
Mixed Finite Elements ◽  
Spatial Dimension ◽  
Symmetric Tensors ◽  
Interior Penalty ◽  
Penalty Term ◽  
The Stability ◽  
Optimal Formula

We propose two classes of mixed finite elements for linear elasticity of any order, with interior penalty for nonconforming symmetric stress approximation. One key point of our method is to introduce some appropriate nonconforming face-bubble spaces based on the local decomposition of discrete symmetric tensors, with which the stability can be easily established. We prove the optimal [Formula: see text]-error estimate for displacement and optimal [Formula: see text] error estimate for stress by adding an interior penalty term. The elements are easy to be implemented thanks to the explicit formulations of its basis functions. Moreover, the method can be applied to arbitrary simplicial grids for any spatial dimension in a unified fashion. Numerical tests for both 2D and 3D are provided to validate our theoretical results.

Memory-type boundary control of a laminated Timoshenko beam

2020 ◽  
Vol 25 (8) ◽  
pp. 1568-1588 ◽  
Author(s):  
Baowei Feng ◽  
Abdelaziz Soufyane
Keyword(s):  
Boundary Conditions ◽  
Timoshenko Beam ◽  
Stability Result ◽  
Kernel Functions ◽  
The Other ◽  
Interfacial Slip ◽  
Small Thickness ◽  
Memory Type ◽  
Type Boundary ◽  
The Stability

In this paper, we consider a laminated Timoshenko beam with boundary conditions of a memory type. This structure is given by two identical uniform layers, one on top of the other, taking into account that an adhesive of small thickness bonds the two surfaces and produces an interfacial slip. Under the assumptions of wider classes of kernel functions, we establish an optimal explicit energy decay result. The stability result is more general than previous works and hence improves earlier results in the literature.

scholarly journals MIXED FINITE ELEMENT APPROXIMATION OF THE VECTOR LAPLACIAN WITH DIRICHLET BOUNDARY CONDITIONS

2012 ◽  
Vol 22 (09) ◽  
pp. 1250024 ◽  
Author(s):  
DOUGLAS N. ARNOLD ◽  
RICHARD S. FALK ◽  
JAY GOPALAKRISHNAN
Keyword(s):  
Finite Element ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Stokes Equations ◽  
Mixed Finite Element ◽  
Mixed Finite Elements ◽  
Mixed Finite Element Method ◽  
Two Dimensions ◽  
Dirichlet Boundary Conditions ◽  
Element Approximation

We consider the finite element solution of the vector Laplace equation on a domain in two dimensions. For various choices of boundary conditions, it is known that a mixed finite element method, in which the rotation of the solution is introduced as a second unknown, is advantageous, and appropriate choices of mixed finite element spaces lead to a stable, optimally convergent discretization. However, the theory that leads to these conclusions does not apply to the case of Dirichlet boundary conditions, in which both components of the solution vanish on the boundary. We show, by computational example, that indeed such mixed finite elements do not perform optimally in this case, and we analyze the suboptimal convergence that does occur. As we indicate, these results have implications for the solution of the biharmonic equation and of the Stokes equations using a mixed formulation involving the vorticity.

Marine ice sheet stability

2012 ◽  
Vol 698 ◽  
pp. 62-72 ◽  
Author(s):  
Christian Schoof
Keyword(s):  
Steady State ◽  
Boundary Conditions ◽  
Multiple Equilibria ◽  
Moving Boundary ◽  
Ice Sheets ◽  
Mass Gain ◽  
Ice Thickness ◽  
Grounding Line ◽  
Type Boundary ◽  
The Stability

AbstractWe examine the stability of two-dimensional marine ice sheets in steady state. The dynamics of marine ice sheets is described by a viscous thin-film model with two Stefan-type boundary conditions at the moving boundary or ‘grounding line’ that marks the transition from grounded to floating ice. One of these boundary conditions constrains ice thickness to be at a local critical value for flotation, which depends on depth to bedrock at the grounding line. The other condition sets ice flux as a function of ice thickness at the grounding line. Depending on the shape of the bedrock, multiple equilibria may be possible. Using a linear stability analysis, we confirm a long-standing heuristic argument that asserts that the stability of these equilibria is determined by a simple mass balance consideration. If an advance in the grounding line away from its steady-state position leads to a net mass gain, the steady state is unstable, and stable otherwise. This also confirms that grounding lines can only be stable in positions where bedrock slopes downwards sufficiently steeply.

Buckling of Circular Cylindrical Shells Using a Displacement Function

1974 ◽  
Vol 96 (4) ◽  
pp. 1322-1327
Author(s):  
Shun Cheng ◽  
C. K. Chang
Keyword(s):  
Boundary Conditions ◽  
Axial Compression ◽  
Cylindrical Shells ◽  
External Pressure ◽  
Displacement Function ◽  
Combined Loading ◽  
Trial Solution ◽  
Governing Differential Equation ◽  
Circular Cylindrical Shells ◽  
The Stability

The buckling problem of circular cylindrical shells under axial compression, external pressure, and torsion is investigated using a displacement function φ. A governing differential equation for the stability of thin cylindrical shells under combined loading of axial compression, external pressure, and torsion is derived. A method for the solutions of this equation is also presented. The advantage in using the present equation over the customary three differential equations for displacements is that only one trial solution is needed in solving the buckling problems as shown in the paper. Four possible combinations of boundary conditions for a simply supported edge are treated. The case of a cylinder under axial compression is carried out in detail. For two types of simple supported boundary conditions, SS1 and SS2, the minimum critical axial buckling stress is found to be 43.5 percent of the well-known classical value Eh/R3(1−ν2) against the 50 percent of the classical value presently known.

scholarly journals Sliding Mode Control and Geometrization Conjecture in Seismic Response

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 353
Author(s):  
Ligia Munteanu ◽  
Dan Dumitriu ◽  
Cornel Brisan ◽  
Mircea Bara ◽  
Veturia Chiroiu ◽  
...  
Keyword(s):  
Sliding Mode Control ◽  
Ricci Flow ◽  
Lyapunov Functions ◽  
Seismic Waves ◽  
Sliding Mode ◽  
Flow Process ◽  
Mode Control ◽  
Finite Period ◽  
The Stability ◽  
Story Building

The purpose of this paper is to study the sliding mode control as a Ricci flow process in the context of a three-story building structure subjected to seismic waves. The stability conditions result from two Lyapunov functions, the first associated with slipping in a finite period of time and the second with convergence of trajectories to the desired state. Simulation results show that the Ricci flow control leads to minimization of the displacements of the floors.

scholarly journals Lidstone–Euler Second-Type Boundary Value Problems: Theoretical and Computational Tools

2021 ◽  
Vol 18 (5) ◽  
Author(s):  
Francesco Aldo Costabile ◽  
Maria Italia Gualtieri ◽  
Anna Napoli
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Existence And Uniqueness ◽  
Interpolation Problem ◽  
Boundary Value ◽  
Computational Tools ◽  
Numerical Examples ◽  
Odd Order ◽  
Uniqueness Of The Solution ◽  
Type Boundary

AbstractGeneral nonlinear high odd-order differential equations with Lidstone–Euler boundary conditions of second type are treated both theoretically and computationally. First, the associated interpolation problem is considered. Then, a theorem of existence and uniqueness of the solution to the Lidstone–Euler second-type boundary value problem is given. Finally, for a numerical solution, two different approaches are illustrated and some numerical examples are included to demonstrate the validity and applicability of the proposed algorithms.

scholarly journals An asymptotically compatible approach for Neumann-type boundary condition on nonlocal problems

2020 ◽  
Vol 54 (4) ◽  
pp. 1373-1413 ◽  
Author(s):  
Huaiqian You ◽  
XinYang Lu ◽  
Nathaniel Task ◽  
Yue Yu
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Nonlocal Boundary ◽  
Nonlocal Diffusion ◽  
Order Convergence ◽  
Flux Boundary Condition ◽  
Nonlocal Interactions ◽  
Local Case ◽  
Neumann Type ◽  
Type Boundary

In this paper we consider 2D nonlocal diffusion models with a finite nonlocal horizon parameter δ characterizing the range of nonlocal interactions, and consider the treatment of Neumann-like boundary conditions that have proven challenging for discretizations of nonlocal models. We propose a new generalization of classical local Neumann conditions by converting the local flux to a correction term in the nonlocal model, which provides an estimate for the nonlocal interactions of each point with points outside the domain. While existing 2D nonlocal flux boundary conditions have been shown to exhibit at most first order convergence to the local counter part as δ → 0, the proposed Neumann-type boundary formulation recovers the local case as O(δ2) in the L∞ (Ω) norm, which is optimal considering the O(δ2) convergence of the nonlocal equation to its local limit away from the boundary. We analyze the application of this new boundary treatment to the nonlocal diffusion problem, and present conditions under which the solution of the nonlocal boundary value problem converges to the solution of the corresponding local Neumann problem as the horizon is reduced. To demonstrate the applicability of this nonlocal flux boundary condition to more complicated scenarios, we extend the approach to less regular domains, numerically verifying that we preserve second-order convergence for non-convex domains with corners. Based on the new formulation for nonlocal boundary condition, we develop an asymptotically compatible meshfree discretization, obtaining a solution to the nonlocal diffusion equation with mixed boundary conditions that converges with O(δ2) convergence.

Low Buckling Loads of Axially Compressed Conical Shells

1970 ◽  
Vol 37 (2) ◽  
pp. 384-392 ◽  
Author(s):  
M. Baruch ◽  
O. Harari ◽  
J. Singer
Keyword(s):  
Boundary Conditions ◽  
Axial Compression ◽  
Plane Boundary ◽  
Essential Difference ◽  
Physical Reason ◽  
Buckling Loads ◽  
Conical Shells ◽  
Simple Supports ◽  
Interaction Curves ◽  
The Stability

The stability of simply supported conical shells under axial compression is investigated for 4 different sets of in-plane boundary conditions with a linear Donnell-type theory. The first two stability equations are solved by the assumed displacement, while the third is solved by a Galerkin procedure. The boundary conditions are satisfied with 4 unknown coefficients in the expression for u and v. Both circumferential and axial restraints are found to be of primary importance. Buckling loads about half the “classical” ones are obtained for all but the stiffest simple supports SS4 (v = u = 0). Except for short shells, the effects do not depend on the length of the shell. The physical reason for the low buckling loads in the SS3 case is explained and the essential difference between cylinder and cone in this case is discussed. Buckling under combined axial compression and external or internal pressure is studied and interaction curves have been calculated for the 4 sets of in-plane boundary conditions.

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