linear statics fem
DESCRIPTION
Linear Statics FEMTRANSCRIPT
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Finite Element Method applied to linear statics ofdeformable solids
Joel Cugnoni, LMAF / EPFL
March 10, 2010
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
1 IntroductionDefinitionsSome MathsLinear elasticity
2 Linear elasticity problemEquilibrium of a linear elastic solidDifferential EquationVirtual Work PrincipleWeak Form
3 The Finite Element MethodApproximate solution of the Weak FormFinite Element Method
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
DefinitionsSome MathsLinear elasticity
Definitions
Linear statics = statics oflinear elastic solids ininfinitesimally smalldeformations
Elastic body =represented in Lagrangianform by its original, stressfree, configuration Ω
Boundary conditions:
Imposed displacementu(x) on δΩu
Imposed surfacetraction t(x) on δΩσ
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
![Page 4: Linear Statics Fem](https://reader034.vdocuments.net/reader034/viewer/2022042721/577c82ea1a28abe054b2cbf0/html5/thumbnails/4.jpg)
OverviewIntroduction
Linear elasticity problemThe Finite Element Method
DefinitionsSome MathsLinear elasticity
Definitions
Linear statics = statics oflinear elastic solids ininfinitesimally smalldeformations
Elastic body =represented in Lagrangianform by its original, stressfree, configuration Ω
Boundary conditions:
Imposed displacementu(x) on δΩu
Imposed surfacetraction t(x) on δΩσ
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
![Page 5: Linear Statics Fem](https://reader034.vdocuments.net/reader034/viewer/2022042721/577c82ea1a28abe054b2cbf0/html5/thumbnails/5.jpg)
OverviewIntroduction
Linear elasticity problemThe Finite Element Method
DefinitionsSome MathsLinear elasticity
Definitions
Linear statics = statics oflinear elastic solids ininfinitesimally smalldeformations
Elastic body =represented in Lagrangianform by its original, stressfree, configuration Ω
Boundary conditions:
Imposed displacementu(x) on δΩu
Imposed surfacetraction t(x) on δΩσ
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
![Page 6: Linear Statics Fem](https://reader034.vdocuments.net/reader034/viewer/2022042721/577c82ea1a28abe054b2cbf0/html5/thumbnails/6.jpg)
OverviewIntroduction
Linear elasticity problemThe Finite Element Method
DefinitionsSome MathsLinear elasticity
Some Maths
Symmetric 2nd order tensors written in vector forms:
σ = σ11, σ22, σ33, σ23, σ31, σ12T
Differential operator:
∇ =
∂/∂x1 0 00 ∂/∂x2 00 0 ∂/∂x3
0 ∂/∂x3 ∂/∂x2
∂/∂x3 0 ∂/∂x1
∂/∂x2 ∂/∂x1 0
hence we have:
div(σ) = ∇Tσ
and:
grad(u) = ∇u
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
DefinitionsSome MathsLinear elasticity
Some Maths
Symmetric 2nd order tensors written in vector forms:
σ = σ11, σ22, σ33, σ23, σ31, σ12T
Differential operator:
∇ =
∂/∂x1 0 00 ∂/∂x2 00 0 ∂/∂x3
0 ∂/∂x3 ∂/∂x2
∂/∂x3 0 ∂/∂x1
∂/∂x2 ∂/∂x1 0
hence we have:
div(σ) = ∇Tσ
and:
grad(u) = ∇u
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
DefinitionsSome MathsLinear elasticity
Linear elasticity
The stress is defined by the symmetric Cauchy stress tensor σ
σ(x) = σ11, σ22, σ33, σ23, σ31, σ12T
For small deformations, the (infinitesimal) strain tensorε(x) = ε11, ε22, ε33, ε23, ε31, ε12T is defined by:
ε(x) = ∇u(x)
The constitutive stress - strain relationship of linear elasticityis given by the Hook’s law:
σ(x) = C(x) ε(x)
where C(x) is the 4th order elasticity tensor
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
DefinitionsSome MathsLinear elasticity
Linear elasticity
The stress is defined by the symmetric Cauchy stress tensor σ
σ(x) = σ11, σ22, σ33, σ23, σ31, σ12T
For small deformations, the (infinitesimal) strain tensorε(x) = ε11, ε22, ε33, ε23, ε31, ε12T is defined by:
ε(x) = ∇u(x)
The constitutive stress - strain relationship of linear elasticityis given by the Hook’s law:
σ(x) = C(x) ε(x)
where C(x) is the 4th order elasticity tensor
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
DefinitionsSome MathsLinear elasticity
Linear elasticity
The stress is defined by the symmetric Cauchy stress tensor σ
σ(x) = σ11, σ22, σ33, σ23, σ31, σ12T
For small deformations, the (infinitesimal) strain tensorε(x) = ε11, ε22, ε33, ε23, ε31, ε12T is defined by:
ε(x) = ∇u(x)
The constitutive stress - strain relationship of linear elasticityis given by the Hook’s law:
σ(x) = C(x) ε(x)
where C(x) is the 4th order elasticity tensor
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
DefinitionsSome MathsLinear elasticity
Elasticity tensor
For an isotropic material in 3D, the 4th order symmetricelasticity tensor is defined by two constants: the Young’smodulus E and the Poisson’s ration ν
With the previous definitions, C can be written in matrix form:
C = ∆
1− ν ν ν 0 0 0ν 1− ν ν 0 0 0ν ν 1− ν 0 0 0
0 0 0 (1−2ν)2 0 0
0 0 0 0 (1−2ν)2 0
0 0 0 0 0 (1−2ν)2
with: ∆ = E
(1+ν)(1−2ν)
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
DefinitionsSome MathsLinear elasticity
Elasticity tensor
For an isotropic material in 3D, the 4th order symmetricelasticity tensor is defined by two constants: the Young’smodulus E and the Poisson’s ration ν
With the previous definitions, C can be written in matrix form:
C = ∆
1− ν ν ν 0 0 0ν 1− ν ν 0 0 0ν ν 1− ν 0 0 0
0 0 0 (1−2ν)2 0 0
0 0 0 0 (1−2ν)2 0
0 0 0 0 0 (1−2ν)2
with: ∆ = E
(1+ν)(1−2ν)
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Equilibrium of a linear elastic solidDifferential EquationVirtual Work PrincipleWeak Form
Stress equilibrium
Internal Stress Equilibrium
Equilibrium is achieved if, atany point x of Ω:
div(σ(x)) + f(x) = 0or
∇Tσ(x) + f(x) = 0
where:
σ is the internal Cauchystress tensor
f is the body (volumic)force vector
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Equilibrium of a linear elastic solidDifferential EquationVirtual Work PrincipleWeak Form
Stress equilibrium
Internal Stress Equilibrium
Equilibrium is achieved if, atany point x of Ω:
div(σ(x)) + f(x) = 0or
∇Tσ(x) + f(x) = 0
where:
σ is the internal Cauchystress tensor
f is the body (volumic)force vector
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
![Page 15: Linear Statics Fem](https://reader034.vdocuments.net/reader034/viewer/2022042721/577c82ea1a28abe054b2cbf0/html5/thumbnails/15.jpg)
OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Equilibrium of a linear elastic solidDifferential EquationVirtual Work PrincipleWeak Form
Stress equilibrium
Internal Stress Equilibrium
Equilibrium is achieved if, atany point x of Ω:
div(σ(x)) + f(x) = 0or
∇Tσ(x) + f(x) = 0
where:
σ is the internal Cauchystress tensor
f is the body (volumic)force vector
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
![Page 16: Linear Statics Fem](https://reader034.vdocuments.net/reader034/viewer/2022042721/577c82ea1a28abe054b2cbf0/html5/thumbnails/16.jpg)
OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Equilibrium of a linear elastic solidDifferential EquationVirtual Work PrincipleWeak Form
Stress equilibrium
Internal Stress Equilibrium
Equilibrium is achieved if, atany point x of Ω:
div(σ(x)) + f(x) = 0or
∇Tσ(x) + f(x) = 0
where:
σ is the internal Cauchystress tensor
f is the body (volumic)force vector
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
![Page 17: Linear Statics Fem](https://reader034.vdocuments.net/reader034/viewer/2022042721/577c82ea1a28abe054b2cbf0/html5/thumbnails/17.jpg)
OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Equilibrium of a linear elastic solidDifferential EquationVirtual Work PrincipleWeak Form
Stress equilibrium
Internal Stress Equilibrium
Equilibrium is achieved if, atany point x of Ω:
div(σ(x)) + f(x) = 0or
∇Tσ(x) + f(x) = 0
where:
σ is the internal Cauchystress tensor
f is the body (volumic)force vector
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
![Page 18: Linear Statics Fem](https://reader034.vdocuments.net/reader034/viewer/2022042721/577c82ea1a28abe054b2cbf0/html5/thumbnails/18.jpg)
OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Equilibrium of a linear elastic solidDifferential EquationVirtual Work PrincipleWeak Form
Differential problem of linear statics
Linear elasticity problem
Using the constitutive relationship in the equlibrium equation, wecan formulate the linear static problem as:
find the displacement field u(x) such that :
∇T (C ∇u) + f = 0 , ∀x ∈ Ω
satisfying the boundary conditions:
imposed displacement:
u(x) = u(x), ∀x ∈ δΩu
imposed surface traction:
NT C ∇u(x) = t(x),∀x ∈ δΩσ
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Equilibrium of a linear elastic solidDifferential EquationVirtual Work PrincipleWeak Form
Differential problem of linear statics
Linear elasticity problem
Using the constitutive relationship in the equlibrium equation, wecan formulate the linear static problem as:
find the displacement field u(x) such that :
∇T (C ∇u) + f = 0 , ∀x ∈ Ω
satisfying the boundary conditions:
imposed displacement:
u(x) = u(x), ∀x ∈ δΩu
imposed surface traction:
NT C ∇u(x) = t(x),∀x ∈ δΩσ
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Equilibrium of a linear elastic solidDifferential EquationVirtual Work PrincipleWeak Form
Virtual Work Principle
Virtual Work Principle
The weak form of the elastostatic problem can be obtained usingthe Virtual Work Principle. Taking the differential equation, wemultiply it by an arbitrary admissible test function δu and integrateit over the domain Ω.∫
Ωδu[∇T C ∇u + f
]dΩ = 0 , ∀ δu
Note that the initial ∀ x of the local equilibrium diff. equation hasbeen replaced by ∀ δu in the integral form.
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Equilibrium of a linear elastic solidDifferential EquationVirtual Work PrincipleWeak Form
Virtual Work Principle, development..
Applying integration by part, and imposing that δu is continuous,differentiable and satisfies the homogeneous form of thedisplacement boundary condition:
δu(x) = 0 , ∀ x ∈ δΩu
We obtain:
−∫
Ω(∇δu)T C∇u dΩ+
∫δΩσ
δuT t d(δΩ)+
∫ΩδuT f dΩ = 0 , ∀ δu
which terms corresponds to the virtual work of internal stresses,surface tractions and body forces.
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Equilibrium of a linear elastic solidDifferential EquationVirtual Work PrincipleWeak Form
Virtual Work Principle, development..
Applying integration by part, and imposing that δu is continuous,differentiable and satisfies the homogeneous form of thedisplacement boundary condition:
δu(x) = 0 , ∀ x ∈ δΩu
We obtain:
−∫
Ω(∇δu)T C∇u dΩ+
∫δΩσ
δuT t d(δΩ)+
∫ΩδuT f dΩ = 0 , ∀ δu
which terms corresponds to the virtual work of internal stresses,surface tractions and body forces.
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Equilibrium of a linear elastic solidDifferential EquationVirtual Work PrincipleWeak Form
Virtual Work Principle, development..
Applying integration by part, and imposing that δu is continuous,differentiable and satisfies the homogeneous form of thedisplacement boundary condition:
δu(x) = 0 , ∀ x ∈ δΩu
We obtain:
−∫
Ω(∇δu)T C∇u dΩ+
∫δΩσ
δuT t d(δΩ)+
∫ΩδuT f dΩ = 0 , ∀ δu
which terms corresponds to the virtual work of internal stresses,surface tractions and body forces.
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Equilibrium of a linear elastic solidDifferential EquationVirtual Work PrincipleWeak Form
Comments on the function space
From the previous assumptions, the function space of thedisplacement field u(x) is :
Ui =
ui (x) | ui (x) ∈ H1(Ω); ui (x) = ui (x), ∀ x ∈ δΩu
, i = (1, 2, 3)
And the function space of the displacement field δu(x) is :
Vi =δui (x) | δui (x) ∈ H1(Ω); δui (x) = 0, ∀ x ∈ δΩu
, i = (1, 2, 3)
where H1 represents the 1st order Sobolev space of functions onΩ satisfying: ∫
Ω(∇u)T∇u + uT u dΩ < ∞
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Equilibrium of a linear elastic solidDifferential EquationVirtual Work PrincipleWeak Form
Comments on the function space
From the previous assumptions, the function space of thedisplacement field u(x) is :
Ui =
ui (x) | ui (x) ∈ H1(Ω); ui (x) = ui (x), ∀ x ∈ δΩu
, i = (1, 2, 3)
And the function space of the displacement field δu(x) is :
Vi =δui (x) | δui (x) ∈ H1(Ω); δui (x) = 0, ∀ x ∈ δΩu
, i = (1, 2, 3)
where H1 represents the 1st order Sobolev space of functions onΩ satisfying: ∫
Ω(∇u)T∇u + uT u dΩ < ∞
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
![Page 26: Linear Statics Fem](https://reader034.vdocuments.net/reader034/viewer/2022042721/577c82ea1a28abe054b2cbf0/html5/thumbnails/26.jpg)
OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Equilibrium of a linear elastic solidDifferential EquationVirtual Work PrincipleWeak Form
Comments on the function space
From the previous assumptions, the function space of thedisplacement field u(x) is :
Ui =
ui (x) | ui (x) ∈ H1(Ω); ui (x) = ui (x), ∀ x ∈ δΩu
, i = (1, 2, 3)
And the function space of the displacement field δu(x) is :
Vi =δui (x) | δui (x) ∈ H1(Ω); δui (x) = 0, ∀ x ∈ δΩu
, i = (1, 2, 3)
where H1 represents the 1st order Sobolev space of functions onΩ satisfying: ∫
Ω(∇u)T∇u + uT u dΩ < ∞
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Equilibrium of a linear elastic solidDifferential EquationVirtual Work PrincipleWeak Form
Weak form of linear statics
Linear elasticity problem in Weak Form
find the displacement field u(x) ∈ U such that :
∫Ω
(∇δu)T C ∇u dΩ =
∫δΩσ
δuT t d(δΩ) +
∫ΩδuT f dΩ ,
∀δu ∈ V
where:
Ui =
ui (x) | ui (x) ∈ H1(Ω); ui (x) = ui (x), ∀ x ∈ δΩu
Vi =
δui (x) | δui (x) ∈ H1(Ω); δui (x) = 0, ∀ x ∈ δΩu
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Equilibrium of a linear elastic solidDifferential EquationVirtual Work PrincipleWeak Form
Weak form of linear statics
Linear elasticity problem in Weak Form
find the displacement field u(x) ∈ U such that :∫Ω
(∇δu)T C ∇u dΩ =
∫δΩσ
δuT t d(δΩ) +
∫ΩδuT f dΩ ,
∀δu ∈ V
where:
Ui =
ui (x) | ui (x) ∈ H1(Ω); ui (x) = ui (x), ∀ x ∈ δΩu
Vi =
δui (x) | δui (x) ∈ H1(Ω); δui (x) = 0, ∀ x ∈ δΩu
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Equilibrium of a linear elastic solidDifferential EquationVirtual Work PrincipleWeak Form
Weak form of linear statics
Linear elasticity problem in Weak Form
find the displacement field u(x) ∈ U such that :∫Ω
(∇δu)T C ∇u dΩ =
∫δΩσ
δuT t d(δΩ) +
∫ΩδuT f dΩ ,
∀δu ∈ V
where:
Ui =
ui (x) | ui (x) ∈ H1(Ω); ui (x) = ui (x), ∀ x ∈ δΩu
Vi =
δui (x) | δui (x) ∈ H1(Ω); δui (x) = 0, ∀ x ∈ δΩu
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Equilibrium of a linear elastic solidDifferential EquationVirtual Work PrincipleWeak Form
Weak form of linear statics
Linear elasticity problem in Weak Form
find the displacement field u(x) ∈ U such that :∫Ω
(∇δu)T C ∇u dΩ =
∫δΩσ
δuT t d(δΩ) +
∫ΩδuT f dΩ ,
∀δu ∈ V
where:
Ui =
ui (x) | ui (x) ∈ H1(Ω); ui (x) = ui (x), ∀ x ∈ δΩu
Vi =
δui (x) | δui (x) ∈ H1(Ω); δui (x) = 0, ∀ x ∈ δΩu
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Equilibrium of a linear elastic solidDifferential EquationVirtual Work PrincipleWeak Form
Comments
The displacement boundary conditions have been moved inthe class of functions U . This type of boundary condition iscalled ”essential” because it is related to the ”essence” of theproblem = the displacement field!!
The surface traction boundary conditions t on δΩσ are nowexplicitly introduced in the weak form. This type of boundarycondition is called ”natural” because it falls ”naturally” in theweak form of the problem!!
The name ”Weak Form” comes from the fact that therequirements of differentiability / continuity of the fields uand δu are relaxed compared to the initial differential equationform. Note that the functions do not necessarilly need to becontinuous!
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
![Page 32: Linear Statics Fem](https://reader034.vdocuments.net/reader034/viewer/2022042721/577c82ea1a28abe054b2cbf0/html5/thumbnails/32.jpg)
OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Equilibrium of a linear elastic solidDifferential EquationVirtual Work PrincipleWeak Form
Comments
The displacement boundary conditions have been moved inthe class of functions U . This type of boundary condition iscalled ”essential” because it is related to the ”essence” of theproblem = the displacement field!!
The surface traction boundary conditions t on δΩσ are nowexplicitly introduced in the weak form. This type of boundarycondition is called ”natural” because it falls ”naturally” in theweak form of the problem!!
The name ”Weak Form” comes from the fact that therequirements of differentiability / continuity of the fields uand δu are relaxed compared to the initial differential equationform. Note that the functions do not necessarilly need to becontinuous!
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
![Page 33: Linear Statics Fem](https://reader034.vdocuments.net/reader034/viewer/2022042721/577c82ea1a28abe054b2cbf0/html5/thumbnails/33.jpg)
OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Equilibrium of a linear elastic solidDifferential EquationVirtual Work PrincipleWeak Form
Comments
The displacement boundary conditions have been moved inthe class of functions U . This type of boundary condition iscalled ”essential” because it is related to the ”essence” of theproblem = the displacement field!!
The surface traction boundary conditions t on δΩσ are nowexplicitly introduced in the weak form. This type of boundarycondition is called ”natural” because it falls ”naturally” in theweak form of the problem!!
The name ”Weak Form” comes from the fact that therequirements of differentiability / continuity of the fields uand δu are relaxed compared to the initial differential equationform. Note that the functions do not necessarilly need to becontinuous!
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
Approximate solution of the Weak Form
We are looking for an approximate solution of the Weak Formproblem. We give us two subsets Uh and Vh of the function spacesU and V to form the basis of the approximations:
uh ≈ u , δuh ≈ δuuh ∈ Uh ⊂ U , δuh ∈ Vh ⊂ V
By considering an approximation Uh based on a linearcombination of a finite number of basis functions, we can write:
uh(x) = H(x) q
where H(x) is the 3xn matrix of shape functions and q is thevector of displacement components. Using Galerkin method, wechoose the same base function space for δu:
δuh(x) = H(x) δq
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
Approximate solution of the Weak Form
We are looking for an approximate solution of the Weak Formproblem. We give us two subsets Uh and Vh of the function spacesU and V to form the basis of the approximations:
uh ≈ u , δuh ≈ δuuh ∈ Uh ⊂ U , δuh ∈ Vh ⊂ V
By considering an approximation Uh based on a linearcombination of a finite number of basis functions, we can write:
uh(x) = H(x) q
where H(x) is the 3xn matrix of shape functions and q is thevector of displacement components.
Using Galerkin method, wechoose the same base function space for δu:
δuh(x) = H(x) δq
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
Approximate solution of the Weak Form
We are looking for an approximate solution of the Weak Formproblem. We give us two subsets Uh and Vh of the function spacesU and V to form the basis of the approximations:
uh ≈ u , δuh ≈ δuuh ∈ Uh ⊂ U , δuh ∈ Vh ⊂ V
By considering an approximation Uh based on a linearcombination of a finite number of basis functions, we can write:
uh(x) = H(x) q
where H(x) is the 3xn matrix of shape functions and q is thevector of displacement components. Using Galerkin method, wechoose the same base function space for δu:
δuh(x) = H(x) δq
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
Approximation of the Weak Form
Using the approximations uh = H q and δuh = H δq, the weakform can be rewritten as:
δqT
∫Ω
(∇H)T C∇H q dΩ−∫
ΩHT f dΩ−
∫δΩσ
HT t d(δΩ)
= 0
This relation must hold ∀δuh ∈ Vh and thus ∀δq, thus:∫Ω
BT CB q dΩ =
∫Ω
HT f dΩ +
∫δΩσ
HT t d(δΩ)
where B = ∇H is called the displacement-strain interpolationmatrix because ε = ∇(Hq) = ∇H q = Bq
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
Approximation of the Weak Form
Using the approximations uh = H q and δuh = H δq, the weakform can be rewritten as:
δqT
∫Ω
(∇H)T C∇H q dΩ−∫
ΩHT f dΩ−
∫δΩσ
HT t d(δΩ)
= 0
This relation must hold ∀δuh ∈ Vh and thus ∀δq, thus:∫Ω
BT CB q dΩ =
∫Ω
HT f dΩ +
∫δΩσ
HT t d(δΩ)
where B = ∇H is called the displacement-strain interpolationmatrix because ε = ∇(Hq) = ∇H q = Bq
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
Approximation of the Weak Form
Using the approximations uh = H q and δuh = H δq, the weakform can be rewritten as:
δqT
∫Ω
(∇H)T C∇H q dΩ−∫
ΩHT f dΩ−
∫δΩσ
HT t d(δΩ)
= 0
This relation must hold ∀δuh ∈ Vh and thus ∀δq, thus:∫Ω
BT CB q dΩ =
∫Ω
HT f dΩ +
∫δΩσ
HT t d(δΩ)
where B = ∇H is called the displacement-strain interpolationmatrix because ε = ∇(Hq) = ∇H q = Bq
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
Approximate linear statics problem
Approximate linear statics problem
find the displacement components q such that :
K q = r
and:uh(x) = H(x) q = u , ∀x ∈ δΩu
with:
K =
∫Ω
BT CB dΩ
r =
∫Ω
HT f dΩ +
∫δΩσ
HT t d(δΩ)
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
Approximate linear statics problem
Approximate linear statics problem
find the displacement components q such that :
K q = r
and:uh(x) = H(x) q = u , ∀x ∈ δΩu
with:
K =
∫Ω
BT CB dΩ
r =
∫Ω
HT f dΩ +
∫δΩσ
HT t d(δΩ)
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
Approximate linear statics problem
Approximate linear statics problem
find the displacement components q such that :
K q = r
and:uh(x) = H(x) q = u , ∀x ∈ δΩu
with:
K =
∫Ω
BT CB dΩ
r =
∫Ω
HT f dΩ +
∫δΩσ
HT t d(δΩ)
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
Comments
K is called the stiffness matrix of the system, r is the forcevector containing contributions of both surface tractions andbody forces
By choosing a finite subset of basis functions to build theapproximation of the weak, we have turned the algebraicdifferential equation problem into a linear system of equations:all terms of K and r are constant, hence the name ”linearstatic problem”.As the displacement is a linear combination of functionsdefined over Ω, enforcing exact displacement boundaryconditions uh(x) = u is not trivial except for zero imposeddisplacement.Integrating the matrices & force vector over an arbitrarydomain can become very complex, and thus numericalintegration methods may be used.
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
Comments
K is called the stiffness matrix of the system, r is the forcevector containing contributions of both surface tractions andbody forcesBy choosing a finite subset of basis functions to build theapproximation of the weak, we have turned the algebraicdifferential equation problem into a linear system of equations:all terms of K and r are constant, hence the name ”linearstatic problem”.
As the displacement is a linear combination of functionsdefined over Ω, enforcing exact displacement boundaryconditions uh(x) = u is not trivial except for zero imposeddisplacement.Integrating the matrices & force vector over an arbitrarydomain can become very complex, and thus numericalintegration methods may be used.
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
![Page 45: Linear Statics Fem](https://reader034.vdocuments.net/reader034/viewer/2022042721/577c82ea1a28abe054b2cbf0/html5/thumbnails/45.jpg)
OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
Comments
K is called the stiffness matrix of the system, r is the forcevector containing contributions of both surface tractions andbody forcesBy choosing a finite subset of basis functions to build theapproximation of the weak, we have turned the algebraicdifferential equation problem into a linear system of equations:all terms of K and r are constant, hence the name ”linearstatic problem”.As the displacement is a linear combination of functionsdefined over Ω, enforcing exact displacement boundaryconditions uh(x) = u is not trivial except for zero imposeddisplacement.
Integrating the matrices & force vector over an arbitrarydomain can become very complex, and thus numericalintegration methods may be used.
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
![Page 46: Linear Statics Fem](https://reader034.vdocuments.net/reader034/viewer/2022042721/577c82ea1a28abe054b2cbf0/html5/thumbnails/46.jpg)
OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
Comments
K is called the stiffness matrix of the system, r is the forcevector containing contributions of both surface tractions andbody forcesBy choosing a finite subset of basis functions to build theapproximation of the weak, we have turned the algebraicdifferential equation problem into a linear system of equations:all terms of K and r are constant, hence the name ”linearstatic problem”.As the displacement is a linear combination of functionsdefined over Ω, enforcing exact displacement boundaryconditions uh(x) = u is not trivial except for zero imposeddisplacement.Integrating the matrices & force vector over an arbitrarydomain can become very complex, and thus numericalintegration methods may be used.
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
The Finite Element Method
In some sense, the Finite Element Method is one way of defining asystematic approximate function space with the aim to:
1 build approximate solutions over arbitrarily complex domains
2 simplify the treatment of essential boundary conditions
3 simplify the integration of the global system matrice(s) andload vector(s)
4 automate the resolution procedure: integration of the linearsystem & solution
Now let’s see how this can be achieved !!
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
The Finite Element Method
In some sense, the Finite Element Method is one way of defining asystematic approximate function space with the aim to:
1 build approximate solutions over arbitrarily complex domains
2 simplify the treatment of essential boundary conditions
3 simplify the integration of the global system matrice(s) andload vector(s)
4 automate the resolution procedure: integration of the linearsystem & solution
Now let’s see how this can be achieved !!
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
![Page 49: Linear Statics Fem](https://reader034.vdocuments.net/reader034/viewer/2022042721/577c82ea1a28abe054b2cbf0/html5/thumbnails/49.jpg)
OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
The Finite Element Method
In some sense, the Finite Element Method is one way of defining asystematic approximate function space with the aim to:
1 build approximate solutions over arbitrarily complex domains
2 simplify the treatment of essential boundary conditions
3 simplify the integration of the global system matrice(s) andload vector(s)
4 automate the resolution procedure: integration of the linearsystem & solution
Now let’s see how this can be achieved !!
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
![Page 50: Linear Statics Fem](https://reader034.vdocuments.net/reader034/viewer/2022042721/577c82ea1a28abe054b2cbf0/html5/thumbnails/50.jpg)
OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
The Finite Element Method
In some sense, the Finite Element Method is one way of defining asystematic approximate function space with the aim to:
1 build approximate solutions over arbitrarily complex domains
2 simplify the treatment of essential boundary conditions
3 simplify the integration of the global system matrice(s) andload vector(s)
4 automate the resolution procedure: integration of the linearsystem & solution
Now let’s see how this can be achieved !!
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
![Page 51: Linear Statics Fem](https://reader034.vdocuments.net/reader034/viewer/2022042721/577c82ea1a28abe054b2cbf0/html5/thumbnails/51.jpg)
OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
The Finite Element Method
In some sense, the Finite Element Method is one way of defining asystematic approximate function space with the aim to:
1 build approximate solutions over arbitrarily complex domains
2 simplify the treatment of essential boundary conditions
3 simplify the integration of the global system matrice(s) andload vector(s)
4 automate the resolution procedure: integration of the linearsystem & solution
Now let’s see how this can be achieved !!
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
The key ideas behind the Finite Element Method
How to process complex domain?
By discretizing the domain in a set of smaller & simpler domains.The discretization Ωh of Ω is defined by a set of discrete pointscalled nodes and a set of sub-domains of basic topology calledelements, which forms what we call a mesh.
Ωh ≈ Ω , Ωh = Ω1 ∪ Ω2 ∪ ... ∪ Ωn
The j-th node is represented by its coordinates xj and the eachelement is defined by its connectivities: an ordered list of nodeswhich serve as support for the element.
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
The key ideas behind the Finite Element Method
How to process complex domain?
By discretizing the domain in a set of smaller & simpler domains.
The discretization Ωh of Ω is defined by a set of discrete pointscalled nodes and a set of sub-domains of basic topology calledelements, which forms what we call a mesh.
Ωh ≈ Ω , Ωh = Ω1 ∪ Ω2 ∪ ... ∪ Ωn
The j-th node is represented by its coordinates xj and the eachelement is defined by its connectivities: an ordered list of nodeswhich serve as support for the element.
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
The key ideas behind the Finite Element Method
How to process complex domain?
By discretizing the domain in a set of smaller & simpler domains.The discretization Ωh of Ω is defined by a set of discrete pointscalled nodes and a set of sub-domains of basic topology calledelements, which forms what we call a mesh.
Ωh ≈ Ω , Ωh = Ω1 ∪ Ω2 ∪ ... ∪ Ωn
The j-th node is represented by its coordinates xj and the eachelement is defined by its connectivities: an ordered list of nodeswhich serve as support for the element.
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
The key ideas behind the Finite Element Method
How to process complex domain?
By discretizing the domain in a set of smaller & simpler domains.The discretization Ωh of Ω is defined by a set of discrete pointscalled nodes and a set of sub-domains of basic topology calledelements, which forms what we call a mesh.
Ωh ≈ Ω , Ωh = Ω1 ∪ Ω2 ∪ ... ∪ Ωn
The j-th node is represented by its coordinates xj and the eachelement is defined by its connectivities: an ordered list of nodeswhich serve as support for the element.
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
The key ideas behind the Finite Element Method
A simple mesh made of 16 elements and 25 nodes
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
The key ideas behind the Finite Element Method
An much more complex mesh of an HydroptereJoel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
The key ideas behind the Finite Element Method
How to simplify the treatment of essential BC?
Use discrete node-based shape functions on Ωh : each node ofcoordinates xi is associated to its own shape function hi . Theapproximation uh is then written as :
uh = H q with H = [h1 I, h2 I, ..., hn I]
and q =
q1,q2, ...,qpT
Moreover, the shape functions are required to have compactsupport (Kronecker property):
hi (xj) = δij ∀xj ∈ Ωh
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
The key ideas behind the Finite Element Method
How to simplify the treatment of essential BC?
Use discrete node-based shape functions on Ωh : each node ofcoordinates xi is associated to its own shape function hi . Theapproximation uh is then written as :
uh = H q with H = [h1 I, h2 I, ..., hn I]
and q =
q1,q2, ...,qpT
Moreover, the shape functions are required to have compactsupport (Kronecker property):
hi (xj) = δij ∀xj ∈ Ωh
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
The key ideas behind the Finite Element Method
An illustration of h5(x) showing the compact support property
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
How to simplify the treatment of essential BC?
Thanks to the Kronecker property, the approximate displacementuh on nodes xj is then:
uh(xj) =
p∑i=1
hi (xj) qi = qj
In other words, the vector q has now the physical meaning: itrepresents the unknown displacements at nodes!Thus the essential boundary conditions can be simply rewritten as:
u(xj) = u(xj) ∀ xj ∈ δΩu =⇒ qj = u(xj) = qj ∀ j | xj ∈ δΩu
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
How to simplify the treatment of essential BC?
Thanks to the Kronecker property, the approximate displacementuh on nodes xj is then:
uh(xj) =
p∑i=1
hi (xj) qi = qj
In other words, the vector q has now the physical meaning: itrepresents the unknown displacements at nodes!Thus the essential boundary conditions can be simply rewritten as:
u(xj) = u(xj) ∀ xj ∈ δΩu =⇒ qj = u(xj) = qj ∀ j | xj ∈ δΩu
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
The key ideas behind the Finite Element Method
How to simplify the integration?
Split integration in each element..∫Ω
dΩ =n∑
i=1
∫Ωi
dΩ
∫δΩ
d(δΩ) =k∑
i=1
∫δΩi
d(δΩ)
Note that local support means that, when integrating hi , onlycontributions of element containing node i must be processed.
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
The key ideas behind the Finite Element Method
How to simplify the integration?
Split integration in each element..∫Ω
dΩ =n∑
i=1
∫Ωi
dΩ
∫δΩ
d(δΩ) =k∑
i=1
∫δΩi
d(δΩ)
Note that local support means that, when integrating hi , onlycontributions of element containing node i must be processed.
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
The key ideas behind the Finite Element Method
How to simplify the integration?∫Ω
dΩ =
∫Ω1
dΩ +
∫Ω2
dΩ +
∫Ω3
dΩ +
∫Ω4
dΩ
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
The key ideas behind the Finite Element Method
How to automate the process?
By treating the problem ”per element” : localization !Local displacement approximation in the e-th element as :
euh = eH eq
Where:eH = [eh1I, eh2I, ..., ehpI]
And:eq = eL q
eL is called the binary localization matrix representing the global -local numbering relationship (related to the element connectivitytable)
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
The key ideas behind the Finite Element Method
How to automate the process?
By treating the problem ”per element” : localization !
Local displacement approximation in the e-th element as :
euh = eH eq
Where:eH = [eh1I, eh2I, ..., ehpI]
And:eq = eL q
eL is called the binary localization matrix representing the global -local numbering relationship (related to the element connectivitytable)
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
The key ideas behind the Finite Element Method
How to automate the process?
By treating the problem ”per element” : localization !Local displacement approximation in the e-th element as :
euh = eH eq
Where:eH = [eh1I, eh2I, ..., ehpI]
And:eq = eL q
eL is called the binary localization matrix representing the global -local numbering relationship (related to the element connectivitytable)
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
The key ideas behind the Finite Element Method
How to automate the process?
By treating the problem ”per element” : localization !Local displacement approximation in the e-th element as :
euh = eH eq
Where:eH = [eh1I, eh2I, ..., ehpI]
And:eq = eL q
eL is called the binary localization matrix representing the global -local numbering relationship (related to the element connectivitytable)
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
The key ideas behind the Finite Element Method
An illustration of the global shape function h5(x)
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
The key ideas behind the Finite Element Method
The local shape functions corresponding to h5(x)
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
The key ideas behind the Finite Element Method
Connectivity table : Local - Global node numbering
eΩ Ω1 Ω2 Ω3 Ω4
1 1 2 4 52 2 3 5 63 5 6 8 94 4 5 7 8
Note:
The connectivity list of each element correspond to a columnof the above table.
The localization operator eL is constructed from theconnectivity table.
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
The key ideas behind the Finite Element Method
Integration of the system of equationUsing the localized quantities euh and eH, we can expand theintegration:
K =
∫Ωh
BT C B dΩ =
q∑e=1
eLT
(∫Ωe
eBT eC eB dΩ
)eL
where eB = ∇ eH is the local displacement - strain matrixand eC is the element elasticity matrix (strain - stressrelationship).
We then define the elementary stiffness matrix eK:
eK =
∫Ωe
eBT eC eB dΩ
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
The key ideas behind the Finite Element Method
Assembly operation
We define the so-called Assembly operator⊎
as:
p⊎e=1
· =
p∑e=1
eLT ·e L
With this operator:
K =
p∑e=1
eLT eKeL =
p⊎e=1
eK
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
The key ideas behind the Finite Element Method
Automating the integration / standardizing the shapefunctions
To automate the integration and simplify the definition of shapefunctions, we transform each ”distorted” elementary domain Ωe
into a reference domain Ωa where we can apply standard numericalintegration procedures.
The coordinate transformation
eT : Ωa → Ωe | ξ → x(ξ)
maps any point of coordinate ξ = ξ1, ξ2, ξ3T in the masterdomain Ωa to its (single) corresponding point of coordinatex = x(ξ) in the elementary domain Ωe (bijective application).
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
The key ideas behind the Finite Element Method
An illustration of the coordinate transform eT
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
The key ideas behind the Finite Element Method
Master element & shape functions
With the coordinate transform eT and eT−1, the shape functionsehi (x) can be mapped to master shape functions ahi (ξ) defined inthe master element space Ωa:
eH(x) = eH(x(ξ)) = aH(ξ)
Such that now, in each element e:
euh[x(ξ)] = aH(ξ) eq
Thanks to that transformation, we only need to derive once theshape function formulations aH(ξ) in the normalized ”master”element space Ωa ⇒ Standardized shape functions !!
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
The key ideas behind the Finite Element Method
Element & Master shape functions
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
The key ideas behind the Finite Element Method
How to choose a simple form of coordinate transformation ?
Simply by using the shape functions to write the coordinatetransform:
eT : x = x(ξ) = aH(ξ) ex
Which means that inside each element e, the coordinates areinterpolated as a linear combination of the shape functions andnodal coordinates ex.With the Kronecker property ehi (x j) = δij ⇔ ahi (ξ
j) = δij , thisensures that each node of the master element corresponds to anode in the ”deformed” element e.
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
The key ideas behind the Finite Element Method
How to choose a simple form of coordinate transformation ?
Simply by using the shape functions to write the coordinatetransform:
eT : x = x(ξ) = aH(ξ) ex
Which means that inside each element e, the coordinates areinterpolated as a linear combination of the shape functions andnodal coordinates ex.With the Kronecker property ehi (x j) = δij ⇔ ahi (ξ
j) = δij , thisensures that each node of the master element corresponds to anode in the ”deformed” element e.
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
The key ideas behind the Finite Element Method
How to choose a simple form of coordinate transformation ?
Simply by using the shape functions to write the coordinatetransform:
eT : x = x(ξ) = aH(ξ) ex
Which means that inside each element e, the coordinates areinterpolated as a linear combination of the shape functions andnodal coordinates ex.With the Kronecker property ehi (x j) = δij ⇔ ahi (ξ
j) = δij , thisensures that each node of the master element corresponds to anode in the ”deformed” element e.
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
The key ideas behind the Finite Element Method
Coordinate transformation
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
The key ideas behind the Finite Element Method
Automating the integration
Using the coordinate transform eT and master shape functions,the integrals can be rewritten to be carried out directly on astandard domain Ωa:
eK =
∫Ωa
( eB(x(ξ))T C eB(x(ξ))
)e j dΩa
Where e j is the determinant of the jacobian matrix eJ
e j = det( eJ)
with the definition:eJij =
∂xi
∂ξj
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
The key ideas behind the Finite Element Method
Master element & derivatives
The spatial derivative operator ∇ is defined in the globalcoordinate system x1, x2, x3. Applying the coordinate transformeT , we can then extend it to be applied on the master element Ωa.Using the standard derivation rules:
∂/∂xi = ∂/∂ξj ∂ξj/∂xi
or in matrix - vector form:
∂/∂x = eJ−1 ∂/∂ξ
With this definition, the elementary strain-displacement matrix eBcan be directly derived from the master shape functions aH andthe integration is completely carried out in the master domain Ωa:
eB = ∇aH(ξ)
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
The key ideas behind the Finite Element Method
Numerical integration
On the master element, standard Gauss-Legendre numericalintegration schemes can be used to automate the calculation of theelementary stiffness matrix eK and load vector. For eachdimension, the integrals are transformed into a weighted sum:∫ +1
−1(·) dξj ≈
r∑i=1
ωij (·) |ξj =ξi
j
where the points of coordinates ξij are called ”Gauss points” or
integration points and ωij are the associated weights.
Using the integration schemes in the three directions:∫ +1
−1
∫ +1
−1
∫ +1
−1(·) dξ3dξ2dξ1 ≈
r∑i=1
s∑j=1
t∑k=1
ωi1ω
j2ω
k3 (·) |
ξ1=ξi1,ξ2=ξj
2,ξ3=ξk3
eK =
∫Ωa
( eB(x(ξ))T C eB(x(ξ))
)e j dΩaJoel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
The key ideas behind the Finite Element Method
Numerical integration
Using the integration schemes in the three directions:∫ +1
−1
∫ +1
−1
∫ +1
−1(·) dξ3dξ2dξ1 ≈
r∑i=1
s∑j=1
t∑k=1
ωi1ω
j2ω
k3 (·) |
ξ1=ξi1,ξ2=ξj
2,ξ3=ξk3
Thus, the elementary stiffness matrix eK can be computednumerically as:
eK ≈r∑
i=1
s∑j=1
t∑k=1
ωi1ω
j2ω
k3
[(∇aH)T C (∇aH) e j
]ξ1=ξi
1,ξ2=ξj2,ξ3=ξk
3
Note that the internal terms (∇aH)T C (∇aH) j are evaluateddirectly in the master element coordinate system ξ1, ξ2, ξ3.
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
The key ideas behind the Finite Element Method
Gauss points & numerical integration
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
The key ideas behind the Finite Element Method
So finally, what is a Finite Element ?
Using these principles, a standard, displacement based, FiniteElement formulation is defined by:
1 The basic topology of its ”master element” Ωa
2 The coordinate transform eT mapping the master domain Ωa
to the real element domain Ωe
3 The shape functions aH and their derivatives ∂hi/∂ξj in themaster element
4 The numerical integration scheme (Gauss points and weights)in the master element.
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
The key ideas behind the Finite Element Method
So finally, what is a Finite Element ?
Using these principles, a standard, displacement based, FiniteElement formulation is defined by:
1 The basic topology of its ”master element” Ωa
2 The coordinate transform eT mapping the master domain Ωa
to the real element domain Ωe
3 The shape functions aH and their derivatives ∂hi/∂ξj in themaster element
4 The numerical integration scheme (Gauss points and weights)in the master element.
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
The key ideas behind the Finite Element Method
So finally, what is a Finite Element ?
Using these principles, a standard, displacement based, FiniteElement formulation is defined by:
1 The basic topology of its ”master element” Ωa
2 The coordinate transform eT mapping the master domain Ωa
to the real element domain Ωe
3 The shape functions aH and their derivatives ∂hi/∂ξj in themaster element
4 The numerical integration scheme (Gauss points and weights)in the master element.
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
![Page 91: Linear Statics Fem](https://reader034.vdocuments.net/reader034/viewer/2022042721/577c82ea1a28abe054b2cbf0/html5/thumbnails/91.jpg)
OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
The key ideas behind the Finite Element Method
So finally, what is a Finite Element ?
Using these principles, a standard, displacement based, FiniteElement formulation is defined by:
1 The basic topology of its ”master element” Ωa
2 The coordinate transform eT mapping the master domain Ωa
to the real element domain Ωe
3 The shape functions aH and their derivatives ∂hi/∂ξj in themaster element
4 The numerical integration scheme (Gauss points and weights)in the master element.
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
![Page 92: Linear Statics Fem](https://reader034.vdocuments.net/reader034/viewer/2022042721/577c82ea1a28abe054b2cbf0/html5/thumbnails/92.jpg)
OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
The key ideas behind the Finite Element Method
So finally, what is a Finite Element ?
Using these principles, a standard, displacement based, FiniteElement formulation is defined by:
1 The basic topology of its ”master element” Ωa
2 The coordinate transform eT mapping the master domain Ωa
to the real element domain Ωe
3 The shape functions aH and their derivatives ∂hi/∂ξj in themaster element
4 The numerical integration scheme (Gauss points and weights)in the master element.
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
![Page 93: Linear Statics Fem](https://reader034.vdocuments.net/reader034/viewer/2022042721/577c82ea1a28abe054b2cbf0/html5/thumbnails/93.jpg)
OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
The key ideas behind the Finite Element Method
So finally, what is a Finite Element Model?
A Finite Element Model is thus defined by :
1 The geometrical mesh Ωh consisting of all elementary domainsΩe (defined by nodes and connectivity table)
2 The definition of the problem to be solved: K q = r
3 A list of Finite Elements (formulations) supported bygeometrical cells of the mesh
4 Boundary conditions: imposed nodal displacement q on δΩu
and imposed surface tractions on δΩσ
5 Material properties ( eC ) assigned to each Finite Element
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
The key ideas behind the Finite Element Method
So finally, what is a Finite Element Model?
A Finite Element Model is thus defined by :
1 The geometrical mesh Ωh consisting of all elementary domainsΩe (defined by nodes and connectivity table)
2 The definition of the problem to be solved: K q = r
3 A list of Finite Elements (formulations) supported bygeometrical cells of the mesh
4 Boundary conditions: imposed nodal displacement q on δΩu
and imposed surface tractions on δΩσ
5 Material properties ( eC ) assigned to each Finite Element
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
The key ideas behind the Finite Element Method
So finally, what is a Finite Element Model?
A Finite Element Model is thus defined by :
1 The geometrical mesh Ωh consisting of all elementary domainsΩe (defined by nodes and connectivity table)
2 The definition of the problem to be solved: K q = r
3 A list of Finite Elements (formulations) supported bygeometrical cells of the mesh
4 Boundary conditions: imposed nodal displacement q on δΩu
and imposed surface tractions on δΩσ
5 Material properties ( eC ) assigned to each Finite Element
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
The key ideas behind the Finite Element Method
So finally, what is a Finite Element Model?
A Finite Element Model is thus defined by :
1 The geometrical mesh Ωh consisting of all elementary domainsΩe (defined by nodes and connectivity table)
2 The definition of the problem to be solved: K q = r
3 A list of Finite Elements (formulations) supported bygeometrical cells of the mesh
4 Boundary conditions: imposed nodal displacement q on δΩu
and imposed surface tractions on δΩσ
5 Material properties ( eC ) assigned to each Finite Element
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
The key ideas behind the Finite Element Method
So finally, what is a Finite Element Model?
A Finite Element Model is thus defined by :
1 The geometrical mesh Ωh consisting of all elementary domainsΩe (defined by nodes and connectivity table)
2 The definition of the problem to be solved: K q = r
3 A list of Finite Elements (formulations) supported bygeometrical cells of the mesh
4 Boundary conditions: imposed nodal displacement q on δΩu
and imposed surface tractions on δΩσ
5 Material properties ( eC ) assigned to each Finite Element
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
The key ideas behind the Finite Element Method
So finally, what is a Finite Element Model?
A Finite Element Model is thus defined by :
1 The geometrical mesh Ωh consisting of all elementary domainsΩe (defined by nodes and connectivity table)
2 The definition of the problem to be solved: K q = r
3 A list of Finite Elements (formulations) supported bygeometrical cells of the mesh
4 Boundary conditions: imposed nodal displacement q on δΩu
and imposed surface tractions on δΩσ
5 Material properties ( eC ) assigned to each Finite Element
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
Shape functions
Requirements
1 Continuity: to have a continuous solution, shape functionsmust be continuous along the interfaces between elements
2 Completeness: to be able to represent uniform strain &displacement, shape functions must at least be affine functions
3 Differentiability: shape functions & derivatives must besufficiently regular to be square integrable over each element
Consequences
Partition of unity: at each point x,∑
i hi (x) = 1
Basic choice of shape function: piecewise polynomials, atleast piecewise trilinear functions in 3D
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
Shape functions
Requirements
1 Continuity: to have a continuous solution, shape functionsmust be continuous along the interfaces between elements
2 Completeness: to be able to represent uniform strain &displacement, shape functions must at least be affine functions
3 Differentiability: shape functions & derivatives must besufficiently regular to be square integrable over each element
Consequences
Partition of unity: at each point x,∑
i hi (x) = 1
Basic choice of shape function: piecewise polynomials, atleast piecewise trilinear functions in 3D
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
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OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
Shape functions
Requirements
1 Continuity: to have a continuous solution, shape functionsmust be continuous along the interfaces between elements
2 Completeness: to be able to represent uniform strain &displacement, shape functions must at least be affine functions
3 Differentiability: shape functions & derivatives must besufficiently regular to be square integrable over each element
Consequences
Partition of unity: at each point x,∑
i hi (x) = 1
Basic choice of shape function: piecewise polynomials, atleast piecewise trilinear functions in 3D
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
![Page 102: Linear Statics Fem](https://reader034.vdocuments.net/reader034/viewer/2022042721/577c82ea1a28abe054b2cbf0/html5/thumbnails/102.jpg)
OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
Shape functions
Requirements
1 Continuity: to have a continuous solution, shape functionsmust be continuous along the interfaces between elements
2 Completeness: to be able to represent uniform strain &displacement, shape functions must at least be affine functions
3 Differentiability: shape functions & derivatives must besufficiently regular to be square integrable over each element
Consequences
Partition of unity: at each point x,∑
i hi (x) = 1
Basic choice of shape function: piecewise polynomials, atleast piecewise trilinear functions in 3D
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
![Page 103: Linear Statics Fem](https://reader034.vdocuments.net/reader034/viewer/2022042721/577c82ea1a28abe054b2cbf0/html5/thumbnails/103.jpg)
OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
Shape functions
Requirements
1 Continuity: to have a continuous solution, shape functionsmust be continuous along the interfaces between elements
2 Completeness: to be able to represent uniform strain &displacement, shape functions must at least be affine functions
3 Differentiability: shape functions & derivatives must besufficiently regular to be square integrable over each element
Consequences
Partition of unity: at each point x,∑
i hi (x) = 1
Basic choice of shape function: piecewise polynomials, atleast piecewise trilinear functions in 3D
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids
![Page 104: Linear Statics Fem](https://reader034.vdocuments.net/reader034/viewer/2022042721/577c82ea1a28abe054b2cbf0/html5/thumbnails/104.jpg)
OverviewIntroduction
Linear elasticity problemThe Finite Element Method
Approximate solution of the Weak FormFinite Element Method
Shape functions
Requirements
1 Continuity: to have a continuous solution, shape functionsmust be continuous along the interfaces between elements
2 Completeness: to be able to represent uniform strain &displacement, shape functions must at least be affine functions
3 Differentiability: shape functions & derivatives must besufficiently regular to be square integrable over each element
Consequences
Partition of unity: at each point x,∑
i hi (x) = 1
Basic choice of shape function: piecewise polynomials, atleast piecewise trilinear functions in 3D
Joel Cugnoni, LMAF / EPFL Finite Element Method applied to linear statics of deformable solids