Introduction to Numerical Methods in Acoustics PDF
Preview Introduction to Numerical Methods in Acoustics
Introduction to Numerical Methods in Acoustics Alfonso Rodriguez-Molares, Ph.D. MEDT 8007 (Day 7) Dept. Circulation and Medical Imaging, Faculty of Medicine, Norwegian University of Science and Technology (NTNU), Trondheim, Norway I have the solution, but it only works in the case of spherical cows in a vacuum 2 Objective → 3 Contents • Finite Differences – Tons of schemes – Errors and stability – PDE into FDE – Boundary conditions – Perfect Matched Layer • K-Wave – What’s under the hood? – Hands on k-Wave – Learn how to work with it 4 Learning by doing • Check out http://folk.ntnu.no/alfonsom/MEDT8007/ • 9 problems • Links & Refs “For the things we have to learn before we can do them, we learn by doing them.” 5 Aristotle, The Nicomachean Ethics A note on notation � �� �� � � = � , = � , = � , � � �� �� �� ��� �� �� �� �� = , , = � , � , � , � � � �� �� �� � � � � � � � � � � � � = + + = � + � + � �� �� �� ��� ��� ��� 6 The thick of it • Definition of the derivative �� � � + ℎ − �(�) � = = lim � �� ℎ �→� 7 The thick of it • Definition of the forward difference scheme �� � � + ℎ − �(�) � = ≈ � �� ℎ 8 Our first three schemes � � + ℎ − �(�) • Forward difference � ≈ � ℎ � � − �(� − ℎ) • Backward difference � ≈ � ℎ � � + ℎ − �(� − ℎ) • Central differences � ≈ � 2ℎ 9 Problem 1 http://folk.ntnu.no/alfonsom/MEDT8007/ 10
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