Continuous finite element methods for Hamiltonian systems

By applying the continuous finite element methods of ordinary differential equations, the linear element methods are proved having second-order pseudo-symplectic scheme and the quadratic element methods are proved having third-order pseudo- symplectic scheme respectively for general Hamiltonian systems, and they both keep energy conservative. The finite element methods are proved to be symplectic as well as energy conservative for linear Hamiltonian systems. The numerical results are in agree-ment with theory.

Numerical Simulation for Moving Contact Line with Continuous Finite Element Schemes

In this paper,we compute a phase field(diffuse interface)model of CahnHilliard type for moving contact line problems governing the motion of isothermal multiphase incompressible fluids.The generalized Navier boundary condition proposed by Qian et al.[1]is adopted here.We discretize model equations using a continuous finite element method in space and a modified midpoint scheme in time.We apply a penalty formulation to the continuity equation which may increase the stability in the pressure variable.Two kinds of immiscible fluids in a pipe and droplet displacement with a moving contact line under the effect of pressure driven shear flow are studied using a relatively coarse grid.We also derive the discrete energy law for the droplet displacement case,which is slightly different due to the boundary conditions.The accuracy and stability of the scheme are validated by examples,results and estimate order.

Continuous Finite Element Subgrid Basis Functions for Discontinuous Galerkin Schemes on Unstructured Polygonal Voronoi Meshes

We propose a new high order accurate nodal discontinuous Galerkin(DG)method for the solution of nonlinear hyperbolic systems of partial differential equations(PDE)on unstructured polygonal Voronoi meshes.Rather than using classical polynomials of degree N inside each element,in our new approach the discrete solution is represented by piecewise continuous polynomials of degree N within each Voronoi element,using a continuous finite element basis defined on a subgrid inside each polygon.We call the resulting subgrid basis an agglomerated finite element(AFE)basis for the DG method on general polygons,since it is obtained by the agglomeration of the finite element basis functions associated with the subgrid triangles.The basis functions on each sub-triangle are defined,as usual,on a universal reference element,hence allowing to compute universal mass,flux and stiffness matrices for the subgrid triangles once and for all in a pre-processing stage for the reference element only.Consequently,the construction of an efficient quadrature-free algorithm is possible,despite the unstructured nature of the computational grid.High order of accuracy in time is achieved thanks to the ADER approach,making use of an element-local space-time Galerkin finite element predictor.The novel schemes are carefully validated against a set of typical benchmark problems for the compressible Euler and Navier-Stokes equations.The numerical results have been checked with reference solutions available in literature and also systematically compared,in terms of computational efficiency and accuracy,with those obtained by the corresponding modal DG version of the scheme.

NUMERICAL SIMULATION OF THE TEMPERATURE FIELD IN A SOLDER BALL UNDER PULSED LASER AND ITS EFFECT ON THE VIBRATION OF THE SOLDER

The research presented here is focused on the vibration condition of a small volume solder solder ball,which is placed on the surface of a soldering pad and is exerted a pulse modulated continuous wave laser heat source. Finite element method is applied to analyzed the temperature field in the solder ball, and experi- ment is conducted to test the vibration. the results show that,that, the temperature field flucturates with the same frequency as that of the laser pulse, which in turn causes a forced vibration of the same frequency in the liquid solder ball.

A Multiscale Method for Damage Analysis of Quasi-Brittle Heterogeneous Materials

A novel multiscale algorithm based on the higher-order continuum at both micro-and macrostructural level is proposed for the consideration of the quasi-brittle damage response of heterogeneous materials.Herein,the microlevel damage is modelled by the degradation of the homogenized stress and tangent stiffness tensors,which are then upscaled to govern the localization at the macrolevel.The C^1 continuity finite element employing a modified case of Mindlin’s form II strain energy density is derived for the softening analysis.To the authors’knowledge,the finite element discretization based on the strain gradient theory is applied for the modeling of damage evolution at the microstructural level for heterogeneous materials for the first time.The advantage of the novel C1 finite element formulation in comparison with the standard finite element discretization in terms of the regularization efficiency as well as the objectivity has been shown.An isotropic damage law is used for the reduction of the constitutive and nonlocal material behaviour,which is necessary for the physically correct description of the localization formation in quasi-brittle materials.The capabilities of the derived finite element to capture the fully developed localization zones are tested on a random representative volume element(RVE)for several different loading cases.By employing the conventional second-order computational homogenization,the microstructural material constitutive response is averaged over the whole RVE area.In order to model the loss of structural integrity when sharp localization is formed across RVE,the specific conditions which detect a completely formed localization zone are developed.A new failure criterion at the microstructural level has been proposed.The derived finite element formulation,as well as the multiscale damage algorithm,are implemented into the finite element program ABAQUS.The capabilities of the presented multiscale scheme to capture the effects of the deformation localization are demonstrated by few benchmark numerical examples.