2. Basic concepts of computations in engineering sciences

Transkript

1 asic concepts of computations in engineering sciences 1 Introduction alculating technical or natural systems requires the following steps for the setup of the model Idealisation of the system to a form that can be calculated onceptual model ormulation of the equations (equations of motion) which describe the idealised model Mathematical model Solving the equations Numerical model alidation and verification of the model Interpretation of the results To begin with two categories of systems will be considered discrete systems continuous systems niversität Stuttgart home/lehre/l-mhs-1-e/inhlt/olien/olie5tex Institut für asserbau Lehrstuhl für Hydromechanik und Hydrosystemmodellierung p1/16

2 asic concepts of computations in engineering sciences The system response of the discrete system can be described directly by a finite number of constitutive equations In principle the exact balance equations (such as the continuity equation momentum equation energy equation ) for a continuous system can be obtained in the same way as for a discrete system However the response of a continuous system is described by the differential equations instead of a system of algebraic equations for the state variables (such as pressure displacement ) The exact solution of a differential equation which fulfills all boundary conditions can only be obtained for relatively simple systems from a discrete idealisation which can then be calculated for a discrete physical system In the lecture we will introduce methods (EM DM M) to reduce a continuous system to a discrete one niversität Stuttgart home/lehre/l-mhs-1-e/inhlt/olien/olie5tex Institut für asserbau Lehrstuhl für Hydromechanik und Hydrosystemmodellierung p/16

3 alculation of discrete systems The fundamental characteristic of a discrete system is that its state can be described directly and with sufficient accuracy from the values of a finite number of state variables Solution steps: 1 Idealisation of the system: The real system is idealised by a combination of elements Equilibrium in the element: The equilibrium conditions or balance equations for each element are established using the state variables ombination of the elements The conditions for the transition between the elements are used in order to establish a system of equilibrium for the unknown state variables 4 Determine the solution: The equilibrium system is solved for the state variables The solution for each element can be found via the fundamental equilibrium conditions or balance equations niversität Stuttgart home/lehre/l-mhs-1-e/inhlt/olien/olie5tex Institut für asserbau Lehrstuhl für Hydromechanik und Hydrosystemmodellierung p/16

4 alculation of discrete systems Example 1: System of heat conduction wall is made of two homogeneous layers which are in contact PSfrag replacements rea I rea II coefficient of heat transmission coefficient of heat transmission coefficient of heat transmission coefficient of heat transmission K K K K ondition for the temperature at the wall e follow the steps 1-4 In the stationary state (no time dependency) the temperature in the wall is determined by the temperatures and at both outer surfaces and the temperature at the interface The ambient temperatures and are supposed to be known niversität Stuttgart home/lehre/l-mhs-1-e/inhlt/olien/olie5tex Institut für asserbau Lehrstuhl für Hydromechanik und Hydrosystemmodellierung p4/16

5 alculation of discrete systems Establish the thermal equilibrium conditions equation for static heat conduction q =total heat flux = area = gradient of temperature in direction of the heat flux = heat conductivity state variables and The law of heat conduction yields the equilibrium conditions - for the left surface - for the left interface - for the right interface - for the right surface niversität Stuttgart home/lehre/l-mhs-1-e/inhlt/olien/olie5tex Institut für asserbau Lehrstuhl für Hydromechanik und Hydrosystemmodellierung p5/16

6 # # alculation of discrete systems In order to obtain the equations for the state variables we use the equilibrium conditions or the continuity equation for the heat flux In matrix form: % &"' "! % &"' "! % &"' # "! The temperatures and are the known boundary conditions They can therefore be transferred to the right hand side niversität Stuttgart Institut für asserbau Lehrstuhl für Hydromechanik und Hydrosystemmodellierung home/lehre/l-mhs-1-e/inhlt/olien/olie5tex p6/16

7 / / / / / / / / / / ( ( ) ( ) ( ) ) ( ( ) ( ) ( ) ) alculation of discrete systems Example : Linear system of elastic springs PSfrag Thereplacements figure shows three inelastic waggons on a horizontal plane which are coupled by a system of linear elastic springs / bekannt / e follow the steps 1-4 e choose as state variables They characterize the response of the system e use the conditions for the combination of the elements in order to establish the equations for the state variables They are given by the static equilibrium conditions: 0 0 niversität Stuttgart home/lehre/l-mhs-1-e/inhlt/olien/olie5tex Institut für asserbau Lehrstuhl für Hydromechanik und Hydrosystemmodellierung p7/16

8 PSfrag replacements alculation of discrete systems ; := ; :6 < 6 ; :6 4 5>6 < 6 Here the upper index in brackets and the lower index indicate the number of the spring and the @ 1 1 PSfrag replacements 0 0 niversität Stuttgart home/lehre/l-mhs-1-e/inhlt/olien/olie5tex Institut für asserbau Lehrstuhl für Hydromechanik und Hydrosystemmodellierung p8/16

9 K G L alculation of 0 0 we can write for ele- ith respect to the components of displacement ment 1: 0 0 and 0 % "! % &"' "! % &"' "! J HI J HI niversität Stuttgart home/lehre/l-mhs-1-e/inhlt/olien/olie5tex Institut für asserbau Lehrstuhl für Hydromechanik und Hydrosystemmodellierung p9/16

10 K 1 0 K G P p10/16 alculation of discrete systems Element : M! % &M' M! % &M' M! J HN L J HN G or: Thus the combination of elements reduces @ % M! % &M' M! % niversität Stuttgart Institut für asserbau Lehrstuhl für Hydromechanik und Hydrosystemmodellierung home/lehre/l-mhs-1-e/inhlt/olien/olie5tex

11 Q T Q alculation of discrete systems It should be mentioned that the coefficients of the stiffness matrix K can also be obtained from QSR where is the stiffness matrix of element i onclusion The analogy between the analysis of temperature heat flow and the displacement force analysis should be noted In both calculations the matrices of coefficients are very similar They can be derived in the same systematic manner niversität Stuttgart home/lehre/l-mhs-1-e/inhlt/olien/olie5tex Institut für asserbau Lehrstuhl für Hydromechanik und Hydrosystemmodellierung p11/16

12 alculation of continous systems Goal: Derivation of differential equations which express the equilibrium conditions or equations of motion for each element the fundamental or material law and the condition for the combination of elements The differential equations have to be valid on the whole domain of the system or the final determination of the system response we have to formulate the boundary conditions to formulate the initial conditions and to establish the equilibrium conditions (or equations of motion) and the law for the material for a typical differential element This yields the system of differential equations for the state variables niversität Stuttgart home/lehre/l-mhs-1-e/inhlt/olien/olie5tex Institut für asserbau Lehrstuhl für Hydromechanik und Hydrosystemmodellierung p1/16

13 niversität Stuttgart Institut für asserbau Lehrstuhl für Hydromechanik und Hydrosystemmodellierung home/lehre/l-mhs-1-e/inhlt/olien/olie5tex 1 Differential formulation Example: Ideal barrage on permeable soil stationary flow Z Z Z Z Z h impermeable retaining wall seepage flow impermeable boundary PSfrag replacements ^ _` a p1/16

14 l l 1 Differential formulation PSfrag replacements dfe cb dfh cg d h d e cg b or a typical element of the side lengths dx and dy () the total amount of inflowing fluid per time step has to equal the total amount of outflowing fluid per time step d e k g j ig d h cg k b j ib b or nm dfe d h g l h dfh d e cb l e niversität Stuttgart home/lehre/l-mhs-1-e/inhlt/olien/olie5tex Institut für asserbau Lehrstuhl für Hydromechanik und Hydrosystemmodellierung p14/16

15 T T o l o l v m u t t sm 1 Differential formulation Making use of Darcys law we can derive the total flow from the total potential (h): p lo l e b lo l h cg ssumption: uniform permeability Inserting (b) in (a) yields: wx y q mr l e l h It should be noted that the same PDE is obtained for: heat flow electrostatic fields niversität Stuttgart home/lehre/l-mhs-1-e/inhlt/olien/olie5tex Institut für asserbau Lehrstuhl für Hydromechanik und Hydrosystemmodellierung p15/16

16 z b z g { 1 Differential formulation oundary conditions impermeable boundary The boundary condition is a streamline permeable boundary oundary condition is the piezometric head h (potential) or velocity v or h essential boundary condition geometric boundary condition Dirichlet boundary condition natural boundary condition load boundary condition Neumann boundary condition zc The differential equation and the boundary conditions determine the static response for the linear flow niversität Stuttgart home/lehre/l-mhs-1-e/inhlt/olien/olie5tex Institut für asserbau Lehrstuhl für Hydromechanik und Hydrosystemmodellierung p16/16