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1414 < head >
15- < title > FEAScript: Finite element method </ title >
15+ < title > FEAScript: Finite Element Method </ title >
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@@ -42,13 +42,13 @@ <h1 class="top">
4242 />
4343 </ a >
4444 </ h1 >
45- < h1 > Finite element method </ h1 >
45+ < h1 > Finite Element Method </ h1 >
4646
4747 < ul id ="menu ">
48- < li > < a href ="#galerkinresiduals "> Galerkin residuals </ a > </ li >
49- < li > < a href ="#boundaryconditions "> Boundary conditions </ a > </ li >
50- < li > < a href ="#isoparametricmapping "> Isoparametric mapping </ a > </ li >
51- < li > < a href ="#residualscomputation "> Residuals computation </ a > </ li >
48+ < li > < a href ="#galerkinresiduals "> Galerkin Residuals </ a > </ li >
49+ < li > < a href ="#boundaryconditions "> Boundary Conditions </ a > </ li >
50+ < li > < a href ="#isoparametricmapping "> Isoparametric Mapping </ a > </ li >
51+ < li > < a href ="#residualscomputation "> Residuals Computation </ a > </ li >
5252 < li > < a href ="#references "> References</ a > </ li >
5353 </ ul >
5454
@@ -58,36 +58,35 @@ <h1>Finite element method</h1>
5858 equation. Basic concepts of the finite element method are described below.
5959 </ p >
6060 < h2 id ="galerkinresiduals ">
61- < a name ="Galerkin residuals "> </ a > Galerkin residuals
61+ < a name ="Galerkin Residuals "> </ a > Galerkin Residuals
6262 </ h2 >
6363 < p >
6464 Consider a typical formulation of a boundary value problem (BVP):
6565 $$\begin{eqnarray} Lu &= f \quad \text{in} \quad D, \\ Bu &= g \quad
6666 \text{on} \quad \partial D. \end{eqnarray} \tag{1}$$ The solution \(u\) is
67- a function of the position vector \(\bar{x}\) at the domain \(D\) which is
68- a subregion of a one-, two- or tree -dimensional Euclidian space. The
67+ a function of the position vector \(\bar{x}\) at the domain \(D\), which
68+ is a subregion of a one-, two- or three -dimensional Euclidean space. The
6969 position vector \(\bar{x}\) is defined as: \( \bar{x} =\sum_{i=1} ^{n} x_i
70- e_i\), where \(e_i\) are the unit vectors and \(n\) the dimension of the
71- Euclidian space. \(L\) is the differential operator and \(B\) is a
70+ e_i\), where \(e_i\) are the unit vectors and \(n\) is the dimension of
71+ the Euclidean space. \(L\) is the differential operator and \(B\) is a
7272 boundary operator defined at the boundary \(\partial D\). The solution
7373 \(u\) can be approximated using a linear combination of basis functions
7474 \(\phi^1, \phi^2, \phi^3, \dots, \phi^N\) such that: \( u(x) = \sum_{j=1}
75- ^{N} u_j \phi^j(x)\), where the number of the basis functions equals to
76- the number of nodes of the computational mesh. The Galerkin method is
77- seeking a solution that zeroes out every one of the following weighted
78- residuals: $$ \begin{aligned} R_i &= \int_{D} (Lu - f) \phi^i \, dS \\ &=
79- \int_{D} \phi^i \left( L\left( \sum_{j=1}^N u_j \phi^j \right) \right) \,
80- dS - \int_{D} f \phi^i \, dS \\ &\quad \text{for} \quad i = 1, 2, \dots,
81- N. \end{aligned} \tag{2} $$ These integrals depend solely on the nodal
82- values \(u_j\), which are the unknowns to be determined. The number of
83- residuals is equal to the number of basis functions, which in turn is
84- equal to the number of nodes, \(N\), in the computational mesh. to the
85- Setting the Galerkin residuals to zero leads to a system of \(N\)
86- algebraic equations with \(N\) unknowns, which can be solved to obtain the
87- approximate solution.
75+ ^{N} u_j \phi^j(x)\), where the number of basis functions equals the
76+ number of nodes of the computational mesh. The Galerkin method seeks a
77+ solution that zeroes out every one of the following weighted residuals: $$
78+ \begin{aligned} R_i &= \int_{D} (Lu - f) \phi^i \, dS \\ &= \int_{D}
79+ \phi^i \left( L\left( \sum_{j=1}^N u_j \phi^j \right) \right) \, dS -
80+ \int_{D} f \phi^i \, dS \\ &\quad \text{for} \quad i = 1, 2, \dots, N.
81+ \end{aligned} \tag{2} $$ These integrals depend solely on the nodal values
82+ \(u_j\), which are the unknowns to be determined. The number of residuals
83+ is equal to the number of basis functions, which in turn is equal to the
84+ number of nodes, \(N\), in the computational mesh. Setting the Galerkin
85+ residuals to zero leads to a system of \(N\) algebraic equations with
86+ \(N\) unknowns, which can be solved to obtain the approximate solution.
8887 </ p >
8988 < h2 id ="boundaryconditions ">
90- < a name ="Boundary conditions "> </ a > Boundary conditions
89+ < a name ="Boundary Conditions "> </ a > Boundary Conditions
9190 </ h2 >
9291 < p >
9392 In general, there are three common types of boundary conditions that can
@@ -130,7 +129,7 @@ <h2 id="boundaryconditions">
130129 </ p >
131130
132131 < h2 id ="isoparametricmapping ">
133- < a name ="Isoparametric mapping "> </ a > Isoparametric mapping
132+ < a name ="Isoparametric Mapping "> </ a > Isoparametric Mapping
134133 </ h2 >
135134 < p >
136135 The basis functions are defined on reference elements: on the unit
@@ -160,39 +159,38 @@ <h2 id="isoparametricmapping">
160159 of \(\bar{x}\): \(\bar{\xi} = \bar{\xi}(\bar{x})\). The basis functions
161160 are then expressed in the coordinate system of \(D\) (the \(\bar{x}\)
162161 system) as \(\phi_i(\bar{\xi}(\bar{x}))\), where \(i = 1, 2, \dots, n_k\).
163- In case where \(D\) is a two-dimensional domain, the Jacobian of the
162+ In the case where \(D\) is a two-dimensional domain, the Jacobian of the
164163 mapping is the matrix: $$ \bar{\bar{J}} = \begin{vmatrix} x_{\xi} &
165164 y_{\xi} \\ x_{\eta} & y_{\eta} \end{vmatrix}, \tag{7}$$ where \( x_{\xi} =
166165 \frac{\partial x}{\partial \xi}\), \( y_{\xi} = \frac{\partial y}{\partial
167166 \xi}\), \( x_{\eta} = \frac{\partial x}{\partial \eta}\), \( y_{\eta} =
168- \frac{\partial y}{\partial \eta}\).
169- </ p >
170- < p >
171- The relation between the \(\bar{x}\)- and \(\bar{\xi}\)-partial
172- derivatives of the basis function can also be calculated as: $$
173- \bar{\bar{J}} \begin{vmatrix} \frac{\partial \phi_i}{\partial x} \\
174- \frac{\partial \phi_i}{\partial y} \end{vmatrix} = \begin{vmatrix}
175- \frac{\partial \phi_i}{\partial \xi} \\ \frac{\partial \phi_i}{\partial
176- \eta} \end{vmatrix}. \tag{8} $$ By solving the above equation, we can
177- calculate the partial derivatives of the basis function with respect to
178- \(x\) and \(y\): $$ \begin{eqnarray} \frac{\partial \phi_i}{\partial x}
179- &=& \frac{1}{\text{det}\bar{\bar{J}}} \left( y_{\eta} \frac{\partial
180- \phi_i}{\partial \xi} - y_{\xi} \frac{\partial \phi_i}{\partial \eta}
181- \right), \\ \frac{\partial \phi_i}{\partial y} &=&
167+ \frac{\partial y}{\partial \eta}\). The relation between the \(\bar{x}\)-
168+ and \(\bar{\xi}\)-partial derivatives of the basis function can also be
169+ calculated as: $$ \bar{\bar{J}} \begin{vmatrix} \frac{\partial
170+ \phi_i}{\partial x} \\ \frac{\partial \phi_i}{\partial y} \end{vmatrix} =
171+ \begin{vmatrix} \frac{\partial \phi_i}{\partial \xi} \\ \frac{\partial
172+ \phi_i}{\partial \eta} \end{vmatrix}. \tag{8} $$ By solving the above
173+ equation, we can calculate the partial derivatives of the basis function
174+ with respect to \(x\) and \(y\): $$ \begin{eqnarray} \frac{\partial
175+ \phi_i}{\partial x} &=& \frac{1}{\text{det}\bar{\bar{J}}} \left( y_{\eta}
176+ \frac{\partial \phi_i}{\partial \xi} - y_{\xi} \frac{\partial
177+ \phi_i}{\partial \eta} \right), \\ \frac{\partial \phi_i}{\partial y} &=&
182178 \frac{1}{\text{det}\bar{\bar{J}}} \left( x_{\xi} \frac{\partial
183179 \phi_i}{\partial \eta} - x_{\eta} \frac{\partial \phi_i}{\partial \xi}
184- \right), \end{eqnarray} \tag{9} $$ where \(\text{det}\bar{\bar{J}}\) is
185- the determinant of the Jacobian: \( \text{det}\bar{\bar{J}} = x_{\xi}
186- y_{\eta} - x_{\eta} y_{\xi}.\) In the case where \(D\) is a
187- one-dimensional domain, Eq. (6) is reduced to: \( x = \sum_{i=1}^{n_k} x_i
188- \phi_i(\xi). \) The Jacobian of the mapping in this case is: \( J =
189- \frac{dx}{d\xi} = x_{\xi}, \) and the partial derivative of the basis
190- function is: \( \frac{d\phi_i}{dx} = \frac{d\phi_i}{d\xi} \Big/ x_{\xi}.
191- \)
180+ \right). \end{eqnarray} \tag{9}$$
181+ </ p >
182+ < p >
183+ where \(\text{det}\bar{\bar{J}}\) is the determinant of the Jacobian: \(
184+ \text{det}\bar{\bar{J}} = x_{\xi} y_{\eta} - x_{\eta} y_{\xi}.\) In the
185+ case where \(D\) is a one-dimensional domain, Eq. (6) is reduced to: \( x
186+ = \sum_{i=1}^{n_k} x_i \phi_i(\xi). \) The Jacobian of the mapping in this
187+ case is: \( J = \frac{dx}{d\xi} = x_{\xi}, \) and the partial derivative
188+ of the basis function is: \( \frac{d\phi_i}{dx} = \frac{d\phi_i}{d\xi}
189+ \Big/ x_{\xi}. \)
192190 </ p >
193191
194192 < h2 id ="residualscomputation ">
195- < a name ="Residuals computation "> </ a > Residuals computation
193+ < a name ="Residuals Computation "> </ a > Residuals Computation
196194 </ h2 >
197195 < p >
198196 A typical representative of the integrals in the Galerkin residuals is the
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