•The experimental investigation of fracture in PMMA (Polymethyl-methacrylate) plate specimens under mode I was carried out.•The stress and strain field distribution under the critical load value was ...calculated using Finite Element Method (FEM).•The stress–strain criterion for notched specimens under tension was proposed.
This paper shows the results of experimental and numerical investigation of fracture in PMMA (Polymethyl-methacrylate) plate specimens under mode I. Samples had two thicknesses (9.7 and 18 mm) and they were weakened by two types of edge-notches: rounded V-notch with radius 0.5 mm and U-notch with radius 10 mm. During tests, critical load value, fracture initiation point and fracture initiation angle were obtained. Additionally, the fracture process was recorded by a high-speed PHANTOM camera. Crack length growth and changes in crack tip velocity over time were analyzed. Stress and strain fields under the critical load value were calculated using Finite Element Method (FEM). Three-dimensional isoparametric elements were used and non-linearity of material and geometry was also included. A stress–strain criterion for notched specimens under tension was proposed.
•New staggered space–time DG schemes for the incompressible Navier–Stokes equations.•Arbitrary high order of accuracy for incompressible fluids in both space and time.•Staggered formulation allows to ...avoid Riemann solvers in several terms.•New space–time pressure-correction algorithm, leading to very sparse pressure system.•Linear system solvable with a matrix-free GMRES algorithm without preconditioner.
In this paper we propose a novel arbitrary high order accurate semi-implicit space–time discontinuous Galerkin method for the solution of the two dimensional incompressible Navier–Stokes equations on staggered unstructured triangular meshes. Isoparametric finite elements are used to take into account curved domain boundaries. The discrete pressure is defined on the primal triangular grid and the discrete velocity field is defined on an edge-based staggered dual grid. While staggered meshes are state of the art in classical finite difference approximations of the incompressible Navier–Stokes equations, their use in the context of high order DG schemes is novel and still quite rare. Formal substitution of the discrete momentum equation on the dual grid into the discrete continuity equation on the primary grid yields a very sparse four-point block system for the scalar pressure, which is conveniently solved with a matrix-free GMRES algorithm. Note that the same space–time DG scheme on a collocated grid would lead to ten non-zero blocks per element, since substituting the discrete velocity into the discrete continuity equation on a collocated mesh would involve also neighbors of neighbors. From numerical experiments we find that our linear system is well-behaved and that the GMRES method converges quickly even without the use of any preconditioner, which is a unique feature in the context of high order implicit DG schemes. A very simple and efficient Picard iteration is then used in order to derive a space–time pressure correction algorithm that achieves also high order of accuracy in time, which is in general a non-trivial task in the context of high order discretizations for the incompressible Navier–Stokes equations. The flexibility and accuracy of high order space–time DG methods on curved unstructured meshes allows to discretize even complex physical domains with very coarse grids in both, space and time. The use of a staggered grid allows to avoid the use of Riemann solvers in several terms of the discrete equations and significantly reduces the total stencil size of the linear system that needs to be solved for the pressure. The proposed method is verified for approximation polynomials of degree up to p=4 in space and time by solving a series of typical numerical test problems and by comparing the obtained numerical results with available exact analytical solutions or other numerical reference data.
To the knowledge of the authors, this is the first time that a space–time discontinuous Galerkin finite element method is presented for the incompressible Navier–Stokes equations on staggered unstructured grids.
In this present investigation, an eight-nodded isoparametric finite element method is used to model the geometry of the stiffened plate based on first-order shear deformation theory. Three nodded ...isoparametric beam elements with four degrees of freedom per node are employed to model the stiffener's geometry. The carbon nanotubes are distributed through the thickness direction of the stiffened plate. The generalized dynamic equilibrium equation is derived from Lagrange's equation of motion using minimum potential energy. The significance of stiffeners addition, stiffeners dimension, aspect ratio, thickness ratio, boundary conditions, fiber volume fraction and temperature on the natural frequency is scrutinized in detail.
The vibrational responses are predicted numerically for the layered shell panel structure with and without cutout under the variable temperature loading and corrugated composite properties. The ...presence of variable cutout shapes (circular/elliptical/square and rectangular) and sizes are modelled via a generic mathematical macro-mechanical model in the framework of the cubic-order kinematic model. Also, the present model includes the variation of composite properties due to the change in environmental conditions, i.e. the temperature-dependent (TD) and -independent (TID) cases. The computational responses are obtained by taking advantages of the isoparametric finite element technique and the Hamilton principle to derive the final governing equation. The total Lagrangian approach is adopted to compute the responses using the specialized computer code prepared in the MATLAB platform. The frequency responses are predicted considering the effect of a cutout, including the environmental variation and compared with previously published eigenvalues. The model versatility is tested over a variety of examples considering the shell configurations (plate, cylindrical, spherical, hyperboloid, and elliptical), the influential cutout parameter (shape, size, and position) and temperature loading including the corrugated composite properties.
Usually detailed impact simulations are based on isoparametric finite element models. For the inclusion in multibody dynamics simulation, e.g., in the framework of the floating frame of reference, a ...previous model reduction is necessary. A precise representation of the geometry is essential for modeling the dynamics of the impact. However, isoparametric finite elements involve the discretization of the geometry. This work tests isogeometric analysis (IGA) models as an alternative approach in the context of impact simulations in flexible multibody systems. Therefore, the adaption of the flexible multibody system procedure to include IGA models is detailed. The use of nonuniform rational basis splines (NURBS) allows the exact representation of the geometry. The degrees of freedom of the flexible body are afterwards reduced to save computation time in the multibody simulation. To capture precise deformations and stresses in the area of contact as well as elastodynamic effects, a large number of global shape functions is required. As test examples, the impact of an elastic sphere on a rigid surface and the impact of a long elastic rod are simulated and compared to reference solutions.
The bending deflections and the corresponding optimal fiber angle sequences of the subsequent layers have been predicted in this article using a hybrid technique. The structural static responses are ...computed numerically via the isoparametric finite element steps in association with Reddy’s higher order mid-plane theory. The final stacking sequences of individual layers are further predicted through two types of soft computing techniques (particle swarm optimization, PSO; teaching–learning-based optimization, TLBO). The responses (deflection and optimal angle) are obtained via a customized computer code (MATLAB) using the current mathematical model in association with two different optimization algorithms. The accuracy of the currently derived higher order hybrid model is established by conducting a few numerical experimentations. The study indicates the superiority of TLBO technique over PSO for any particular problem when compared to the minimum deflection constraint whereas not many deviations for stacking sequences. Finally, the influences of the different structural parameter are explored by solving a variety of numerical examples and the corresponding inferences provided in detail.
A finite-element scheme based on a coupled arbitrary Lagrangian–Eulerian and Lagrangian approach is developed for the computation of interface flows with soluble surfactants. The numerical scheme is ...designed to solve the time-dependent Navier–Stokes equations and an evolution equation for the surfactant concentration in the bulk phase, and simultaneously, an evolution equation for the surfactant concentration on the interface. Second-order isoparametric finite elements on moving meshes and second-order isoparametric surface finite elements are used to solve these equations. The interface-resolved moving meshes allow the accurate incorporation of surface forces, Marangoni forces and jumps in the material parameters. The lower-dimensional finite-element meshes for solving the surface evolution equation are part of the interface-resolved moving meshes. The numerical scheme is validated for problems with known analytical solutions. A number of computations to study the influence of the surfactants in 3D-axisymmetric rising bubbles have been performed. The proposed scheme shows excellent conservation of fluid mass and of the total mass of the surfactant.
In this article, two types of higher-order kinematic theories are adopted to evaluate the nonlinear bending and the stress values of the internally damaged layered composite flat panel structure ...numerically including the thickness stretching effect. The structural distortion is modeled by Green–Lagrange strain kinematics including all of the nonlinear higher-order strain terms to maintain the required generality. Additionally, the internal debonding between the adjacent layers is introduced via two sub-laminate approaches by maintaining the intermittent link as a priori by the continuity condition. Subsequently, the static equilibrium equations of the debonded structure under the influence of uniform mechanical loading are obtained using a variational principle and solved iteratively in association with the isoparametric finite element steps. Further, the accuracy of the derived model is established by comparing the deflection and stress values with available published results including own experimental data (three-point bend test on artificially debonded layered composite). Finally, a suitable number of numerical examples is solved using the derived higher-order nonlinear models to reveal the operational strength and effect of the debonding (size, position, and location) on the nonlinear static deflection values of the debonded structure.
We present an isoparametric unfitted finite element approach of the CutFEM or Nitsche-XFEM family for the simulation of two-phase Stokes problems with slip between phases. For the unfitted ...generalized Taylor–Hood finite element pair
P
k
+
1
-
P
k
,
k
≥
1
, we show an inf-sup stability property with a stability constant that is independent of the viscosity ratio, slip coefficient, position of the interface with respect to the background mesh and, of course, mesh size. In addition, we prove stability and optimal error estimates that follow from this inf-sup property. We provide numerical results in two and three dimensions to corroborate the theoretical findings and demonstrate the robustness of our approach with respect to the contrast in viscosity, slip coefficient value, and position of the interface relative to the fixed computational mesh.
Among a few known techniques the isoparametric version of the finite element method for meshes consisting of curved triangles or tetrahedra is the one most widely employed to solve PDEs with ...essential conditions prescribed on curved boundaries. It allows to recover optimal approximation properties that hold for elements of order greater than one in the energy norm for polytopic domains. However, besides a geometric complexity, this technique requires the manipulation of rational functions and the use of numerical integration. We consider a simple alternative to deal with Dirichlet boundary conditions that bypasses these drawbacks, without eroding qualitative approximation properties. In the present work we first recall the main principle this technique is based upon, by taking as a model the solution of the Poisson equation with quadratic Lagrange finite elements. Then we show that it extends very naturally to viscous incompressible flow problems. Although the technique applies to any higher order velocity–pressure pairing, as an illustration a thorough study thereof is conducted in the framework of the Stokes system solved by the classical Taylor–Hood method.
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