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Computational structural analysis of dental implants using radial point interpolation meshless methods
Published in J. Belinha, R.M. Natal Jorge, J.C. Reis Campos, Mário A.P. Vaz, João Manuel, R.S. Tavares, Biodental Engineering V, 2019
C.C.C. Coelho, J. Belinha, R.M. Natal Jorge
Depending on the formulation used, meshless methods can be classified into two categories. The first is the strong formulation, that directly uses the partial differential equations that govern the physical phenomenon studied to obtain the solution (Belinha, 2014). Smooth Particle Hydrodynamics (SPH) was the first meshless method to be proposed in this category, as mentioned in (Tavares et al., 2015), being responsible for the origin of the Reproducing Kernel Particle Method (RKPM). A parallel pathway on the development of meshless methods began in the 1990s. This alternative pathway used weak formulation. The weak formulation uses a variable principle to minimize the residual weight of the differential equations that govern the phenomenon. This residue is obtained by replacing the exact solution with an approximate function affected by a test function (Belinha, 2014). The first mature meshless method to be used with this formulation in computational mechanics was the Diffuse Element Method (DEM) (Belinha, 2014), (Tavares et al., 2015) based on the approximation method, the least squares moving (Lancaster & Salkauskas, 1981).
Numerical Methods—the Enthalpy Formulation
Published in Vasilios Alexiades, Alan D. Solomon, Mathematical Modeling of Melting and Freezing Processes, 2018
Vasilios Alexiades, Alan D. Solomon
This theorem justifies the assertion that a weak solution is indeed a generalized concept of solution, capable of existing even when the classical solution may not exist at all. In addition, a weak solution satisfies the equation Ht = ∇2u in the sense of distributions in G. Note that in this formulation there is absolutely no assumption whatsoever regarding the nature of the interface; in fact the interface is not even mentioned ! It is only recovered a posteriori as the set of points (x→,t) where H lies between 0 and ρL. It may happen to be a sharp, smooth surface (in which case the weak solution is in fact a classical solution), or it may be an extended mushy zone. The great advantage of the weak formulation is that we do not need to know the answer in order to state the problem (which is the case with the classical formulation)!
Finite Element Method
Published in Young W. Kwon, Multiphysics and Multiscale Modeling, 2015
in which Γe denotes an element boundary. The first expression of Equation 2.37 is called the strong formulation, and the second expression is called the weak formulation. The Galerkin method results in the test function such that () w1=du˜dui=Hi(x) and w2=du˜duj=Hj(x)
Guaranteed convergence for a class of coupled-cluster methods based on Arponen's extended theory
Published in Molecular Physics, 2020
Simen Kvaal, Andre Laestadius, Tilmann Bodenstein
As is common in analysis of partial differential equations [29,30], we pass to a weak formulation, which in this case is equivalent to the strong formulation outlined above. Under the assumption that is below bounded, we can introduce a unique extension (dual space), where is a dense subspace, a Hilbert space with norm , continuously embedded in . It follows that is continuously embedded in , and we have a scale of spaces with dense embeddings, . The operator H is bounded (i.e. continuous), and satisfies a Gårding estimate, i.e. for some and some , for all . For the electronic-structure problem can be taken to be the space of square-integrable functions with finite kinetic energy.
Study of deformation due to thermal shock in porous thermoelastic material with reference temperature dependent properties
Published in Mechanics Based Design of Structures and Machines, 2022
To obtain the weak formulation, the governing equations are multiplied by independent weighting functions and then integrated over the specified domain. Applying the integration by parts and making use of the Gauss divergence theorem to reduce the order of special derivatives and allows for the application of the boundary conditions. In the framework of the standard Galerkin’s procedure, the weighting functions and shape functions coincide. The displacement component (u), volume fraction field and temperature (T) are related to the corresponding nodal values by where n denotes the number of nodes per element, and N represents the shape functions. For this study, we use eight-/nine-noded isoparametric quadrilateral element, see Figure 1. System of finite element equations corresponding to Equations (13–16) can be obtained as follows: where the summation sign represents the assembly of vectors (not the addition of vectors) in which only the elements corresponding to particular degree of freedom in the different vectors are added. and are the element stiffness, mass, and damping matrices, respectively. is the element vector of nodal forces, and m is the number of elements. The vectors and matrices of an element are evaluated by the following sub-vectors and sub-matrices: where and with where n is the number of nodes on an element. Ni and Nj are the shape functions of the nodes i and j, respectively; and and are the Cauchy surface traction boundary conditions on Γ; nx and ny are direction cosines between the normal and the x and y directions, respectively; Ω is the domain; and Γ is the boundary of the physical domain. qx and qy represent heat flux in x and y directions. For discretization of the present problem, we are using 2D quadratic elements. For 2D figures, we are using eight nodded isoparametric finite element as follows: