Extended finite element and meshfree methods: 12' Smeared cracks, embedded elements, interface eleme

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Extended finite element and meshfree methods12.
Smeared cracks, embedded elements, interface
elements Timon RabczukProf. Wolfgang Wall
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Outline
A continuous SPH-approximation in 1D is given by
(1)
where W is the kernel function given by

• Give the discrete form of (1) using nodal
  integration!
• Derive the discrete form for a gradient of the
  function u using nodal integration and Gauss
  quadrature!
• What is the order of continuity of (1)? Is (1)
  zero- and first order complete (in its continuous
  form)? After discretization (nodal integration),
  is (1) zero and linear complete? Give
  explanations!
• Modify the discrete SPH form (2) such that it is
  at least zero-order complete!

(2)
Vquad. weight
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Outline

• Give the constant and linear reproducing
  conditions and the constant and linear derivative
  reproducing conditions for a two-dimensional
  problem
• Show in the absence of external and body forces
  that a zero-order complete approximation
  guarantees the conservation of linear momentum!

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Motivation
The equilibrium equation in variational
formulation is given by find u such that

• Derive the stiffness matrix and the external
  force vector for the element-free (EFG) (Bubnov)
  Galerkin method! The integrals do not need to be
  evaluated.
• Modify the variational principle (1) such that
  Dirichlet boundary conditions are enforced with
  the Lagrange multiplier method. Give the final
  system of equations (without evaluating the
  integrals)!
• Give the external force vector for a nodal,
  stress-point and Gauss integration!

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Outline

• Figures 1-4 show different crack patterns in an
  XFEM discretization with linear or bilinear shape
  functions. In figures 1-3, only a step enrichment
  is employed in the XFEM formulation. In figure 4,
  the branch enrichment is used in addition to the
  step enrichment to guarantee crack closure at the
  crack tip. The crack is described by different
  level set functions according to figures 1-4.
• Draw all enriched nodes into the figures such
  that crack closure at the crack tip is ensured.
  If several enrichments per node are needed,
  indicate this clearly.
• Give the approximation of the displacement field
  for the elements 1,2 and 3!

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Outline
A discretization of the (linear) Timoshenko beam
problem is given in figure 1. The element-free
Galerkin (EFG-Bubnov Galerkin) with background
integration is used where the integration cells
are created by the EFG nodes as shown in figure
1. Linear finite element shape functions are used
for the background cells. In the EFG method, a
linear polynomial basis p1 x y is used. The
kernel function is the cubic B-Spline with
circular support size. Assume that the
connectivity arrays of the background cells, the
local and global coordinatesthe weights of the
quadrature points, the Jacobian and its
determinant, the support size, the elasticity
matrix and the global position of the EFG nodes
are already known. Write a detailed algorithm how
to assemble the global stiffness matrix.