THESE de DOCTORAT - cerfacs
THESE de DOCTORAT - cerfacs THESE de DOCTORAT - cerfacs
48 Chapter 3: Development of a numerical tool for combustion noise analysis, AVSP-f 3. solve Rx = w for the unknown x. Both of these methods are suitable when dealing with small matrices (n
3.3 Solving the system Ax = b 49 where ̂Q m+1 ∈ C n×m+1 is an orthonormal matrix and ˜H m ∈ C m+1×m is an upper-Hessenberg matrix 4 . ̂D y ˜H y ̂Q m+1 m m +1× m m n × m n × m +1 Figure 3.6: Hessenberg decomposition The least square problem is now given by Find y such that ‖r 0 − Q m+1 ˜H m y‖ 2 = ‖r m ‖ 2 is minimized (3.21) Let us introduce r 0 = q 1 ‖r 0 ‖ (see Eq. 3.43). The vector q 1 can be expressed as Q m+1 e 1 = q 1 where e 1 is the canonical unit vector e 1 = (1, 0, 0, ...). Furthermore, the invariance of inner products means that the angles between vectors are preserved, and so are their lengths: ‖Qx‖ = ‖x‖. The final expression results in ‖Q m+1 ( ‖r0 ‖e 1 − ˜H m y ) ‖ 2 = ‖ ( ‖r‖e 1 − ˜H m y ) ‖ 2 (3.22) leading to Find y such that ‖ ( ‖r‖e 1 − ˜H m y ) ‖ 2 = ‖r m ‖ 2 is minimized (3.23) Once y is found through the least square methodology, a solution for the Ax m ≈ b problem can be found by solving x m = x 0 + ̂Q m y. There are still two crucial issues to resolve: 1. How to construct such matrices ̂Q m and ˜H m ? 2. What is the ‘least square problem’ about ? These two issues will be addressed in sections (3.3.1) and (3.3.2) respectively. 4 An upper- Hessenberg matrix H is a matrix with zeros below the first subdiagonal
- Page 8 and 9: Y gracias familia mía!!, que así
- Page 11: Abstract Today, much of the current
- Page 14 and 15: 4 CONTENTS 3.4.1 Preconditioning .
- Page 16 and 17: List of Figures 1.1 Main sources of
- Page 18 and 19: 8 LIST OF FIGURES 5.19 Acoustic ene
- Page 20: List of Tables 1.1 EU recommended r
- Page 24 and 25: 14 Chapter 1: General Introduction
- Page 26 and 27: 16 Chapter 1: General Introduction
- Page 28 and 29: 18 Chapter 1: General Introduction
- Page 30 and 31: 2 Computation of noise generated by
- Page 32 and 33: 22 Chapter 2: Computation of noise
- Page 34 and 35: 24 Chapter 2: Computation of noise
- Page 36 and 37: 26 Chapter 2: Computation of noise
- Page 38 and 39: 28 Chapter 2: Computation of noise
- Page 40 and 41: 30 Chapter 2: Computation of noise
- Page 42 and 43: 32 Chapter 2: Computation of noise
- Page 44 and 45: 34 Chapter 2: Computation of noise
- Page 46 and 47: 36 Chapter 2: Computation of noise
- Page 48 and 49: 38 Chapter 2: Computation of noise
- Page 50 and 51: 40 Chapter 2: Computation of noise
- Page 52 and 53: 42 Chapter 3: Development of a nume
- Page 54 and 55: 44 Chapter 3: Development of a nume
- Page 56 and 57: 46 Chapter 3: Development of a nume
- Page 60 and 61: 50 Chapter 3: Development of a nume
- Page 62 and 63: 52 Chapter 3: Development of a nume
- Page 64 and 65: 54 Chapter 3: Development of a nume
- Page 66 and 67: 56 Chapter 3: Development of a nume
- Page 68 and 69: 4 Validation of the acoustic code A
- Page 70 and 71: 60 Chapter 4: Validation of the aco
- Page 72 and 73: 62 Chapter 4: Validation of the aco
- Page 74 and 75: 64 Chapter 4: Validation of the aco
- Page 76 and 77: 66 Chapter 4: Validation of the aco
- Page 78 and 79: 68 Chapter 4: Validation of the aco
- Page 80 and 81: 70 Chapter 4: Validation of the aco
- Page 82 and 83: 5 Assessment of combustion noise in
- Page 84 and 85: 74 Chapter 5: Assessment of combust
- Page 86 and 87: 76 Chapter 5: Assessment of combust
- Page 88 and 89: 78 Chapter 5: Assessment of combust
- Page 90 and 91: 80 Chapter 5: Assessment of combust
- Page 92 and 93: 82 Chapter 5: Assessment of combust
- Page 94 and 95: 84 Chapter 5: Assessment of combust
- Page 96 and 97: 86 Chapter 5: Assessment of combust
- Page 98 and 99: 88 Chapter 5: Assessment of combust
- Page 100 and 101: 90 Chapter 5: Assessment of combust
- Page 102 and 103: 92 Chapter 5: Assessment of combust
- Page 104 and 105: 94 Chapter 5: Assessment of combust
- Page 106 and 107: 96 Chapter 6: Boundary conditions f
48 Chapter 3: Development of a numerical tool for combustion noise analysis, AVSP-f<br />
3. solve Rx = w for the unknown x.<br />
Both of these methods are suitable when <strong>de</strong>aling with small matrices (n
Change language