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NUMERICAL METHODS THAT CAN BE USED IN BIOMECHANICS. Mechanics of Materials Approach (A) Complex Beam Theory (i) Straight Beam (ii) Curved Beam (iii) Composite Beam. From:Daviddarling.info. NUMERICAL METHODS THAT CAN BE USED IN BIOMECHANICS. Mechanics of Material Approach (Cont).

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## NUMERICAL METHODS THAT CAN BE USED IN BIOMECHANICS

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**NUMERICAL METHODS THAT CAN BE USED IN BIOMECHANICS**• Mechanics of Materials Approach (A) Complex Beam Theory (i) Straight Beam (ii) Curved Beam (iii) Composite Beam From:Daviddarling.info**NUMERICAL METHODS THAT CAN BE USED IN BIOMECHANICS**Mechanics of Material Approach (Cont)**NUMERICAL METHODS THAT CAN BE USED IN BIOMECHANICS**(2) Finite Difference Method**NUMERICAL METHODS THAT CAN BE USED IN BIOMECHANICS**(2) Finite Difference Method (Contd) Consider an ordinary differential equation One of the difference equation method is using: To approximate the differential equation. Solution is:**Introduction**• Re-invented around 1963 • Initially applied to engineering structures Concrete dams Aircraft structures (Civil engineers) (Aeronautical engineers)**Introduction**• FEM is based on Energy Method Method of Residuals**Introduction**• Energy method Total potential energy must be stationary δ (U + W) = δ ( П ) = 0**Introduction**• Residual method Differential equation governing the problem is given by A ( ø ) = 0 Minimise R = A ( ø* ) - A ( ø ) ø is actual solution ø* is assumed solution**Introduction**• Both methods give us a set of equations [ K ] { a } = { f } Stiffness Matrix Force Matrix Displacement Matrix**Introduction - FEM Procedure**• Continuum is separated by imaginary lines or surfaces into a number of “finite elements” Finite Elements**Introduction - FEM Procedure**• Elements are assumed to be interconnected at a discrete number of “nodal points” situated on their boundaries Nodal Points Finite Elements Displacements at these nodal points will be the basic unknown**Introduction - FEM Procedure**• A set of functions is chosen to define uniquely the state of displacement within each finite element ( U ) in terms of nodal displacements ( a1 , a2 , a3 ) Finite Element a2 Nodal Point U = Σ Ni ai i= 1, 3 y a3 a1 x**Introduction - FEM Procedure**• This displacement function is input into either “energy equations” or “residual equations” to give us element equilibrium equation • [ K ] { a } = { f } Finite Element a2 Nodal Point y Element Displacement Matrix Element Force Matrix Element Stiffness Matrix a3 a1 x**Introduction - FEM Procedure**• Element equilibrium equations are assembled taking care of displacement compatibility at the connecting nodes to give a set of equations that represents equilibrium of the entire continuum Nodal Points Finite Elements**Introduction - FEM Procedure**• Solution for displacements are obtained after substituting boundary conditions in the continuum equilibrium equations Nodal Points Finite Elements Support Points Support Points**Introduction**• Finite element method used to solve: • Elastic continuum • Heat conduction • Electric & Magnetic potential • Non-linear (Material & Geometric) -plasticity, creep • Vibration • Transient problems • Flow of fluids • Combination of above problems • Fracture mechanics**Introduction**• Finite elements: • Truss , Cable and Beam elements • Two & Three solid elements • Axi-symmetric elements • Plate & Shell elements • Spring, Damper & Mass elements • Fluid elements**Finite Element Mesh of C4-C7**Facet Joints C4 C5-C6 Graft C5 C6 C7 Intact With Graft at C5-C6 Level**von Mises Stress in C4-C5 Annulus (Flexion)**5 MPa Anterior 6 MPa Anterior Kyphotic Graft Neutral Graft**Vertical Displacement Distribution**in L1-S1**Finite Element Mesh of L2-L5**With 25% Translational Spondylolisthesis**Vertical Displacement Distribution**in L2-L5 Under Flexion Moment (25% translational spondylolisthesis)**Finite Element Mesh to Represent Tibial Insert & Femoral**Component**Motion of Femoral Implant with respect to UHMWPE Knee Insert****SIGMA-ZZ in cortical bone in a femur with implant attached**using cement**Advantage of using FEM**• Irregular complex geometry can be modeled • Effect of large number of variables in a problem can be easily analysed • Multiple phase problems can be modeled • Effect of various surgical techniques can be compared using appropriate FE models • Both static and time dependent problems can be modeled • Solution to certain problems that cannot be (or difficult) obtained otherwise can be solved by FEM

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